I was working on point by point rebuttals, but it became lengthy so I'll just point to a few things.
Your math isn't wrong but how you're applying it is flawed. There's an inherent conflict between applying the law of large numbers and ignoring those same large numbers outside of the very limited number of uses of tactical mind from second wind. It's not practical in play and the swing of both dice makes that chance erratic in the 2 times the fighter can use it before taking a short rest. In practical play, on the other hand, it's very easy for a player to use skills with expertise dozens of times in a hour if that player wishes.
I am not ignoring it and they are not "very limited" because the resource is only used on a sucess.
The ability to take an action that triggers the use of a skill the player has chosen in which to become an expert is within the agency of the player.
So stealth is a bit misleading because a Rogue at 2nd level can use a bonus action to hide. So they can and do use it regularly in combat without using an action.
You are talking about 1 skill out of 18 here, so saying the Rogue is "better at skills" is not nearly the same as saying the Rogue is better at stealth because he can do it as a bonus action. I will concede that a Rogue is better at hiding specifically than a Fighter and he is better regardless of proficiency and largely regardless of dexterity due to the different action cost, but that is different than being better at skills.
Further, if as you say actually look at only actions specifically; only when the Rogue uses an action to hide compared to when a fighter uses an action to hide (all the time for the fighter). I think it is uncommon that a Rogue will do an action stealth check enough times for the steady state +4 to be mathematically more effective than the +1-10 with no use consumed unless it turns successs into a fail. At 2nd level, assuming a 16 Dex on both PCs and no short rests you need to take more than 5 actions to hide to do better than the fighter on the day, I don't think that is very common and that number goes up at 4th level and it goes up with any short rests.
If I take expertise in persuasion I'm going to try and use that skill as often as possible, which is why I took it. If I take... etc.
With expertise in persuasion, although you may try, I don't think you will commonly use it enough to actually outrun a fighter in play
Also this works both ways - With tactical mind I am going to try a lot of checks in all 18 skills because I took the fighter class.
15 DC is the most common DC and good to bench against, and players playing to their chosen skills with that player agency having +3 ability score bonus and +2 proficiency bonus against that DC is 10% lower than the same bonus with expertise, which is in turn 17% lower than the fighter using the tactical mind bonus.
Tactical mind is only a 7% better bonus over expertise than expertise is over proficiency at those levels under those assumptions even when we do accept those averages.
And expertise is only a 10% bonus over proficiency. Moreover that 7% comes into play on every check until all the uses of Tactical Mind are used up and at DC 15 that is going to be 4 checks on average to burn one use. The expected check is the 8th skill check that the fighter rolls that will use up his 2nd Tactical Mind (with a DC15). So the most common outcome is that there are 8 checks the fighter was 7% ahead of the Rogue for each of those checks (i.e. 8 times)
Now when considering who is better for the day, you have to include the chance that the Fighter will actually use TM quicker, because while a Fighter is 7% better until all TM uses are used up, in fact when looking at it statistically it is not correct to bias the Fighter 7% for 8 straight checks because 7% assumes a use is available, which there is a chance it is not available starting at check 3. So it is 7% for the first two and then it goes down from there.
What you end up with is 5 checks is the crossover point. The 5th check is statistically where the Rogue starts doing better than the fighter overall. That is assuming expertise for the Rogue and assuming no short rests.
Bards, ranger, and rogues not only have expertise but also an extra skill proficiency or two over the fighter to apply.
No they don't
In tier 1 Rogues and Fighters are both proficient in 4 skills. Rogues additionally have expertise in 2 skills.
I am focusing on the skills with expertise because in the other 16 skills they are exactly the same (or 17 in the case of Ranger).
And yes, a fighter can use TM on any of those 16 skills that they are the same in, or on an ability check that is not a skill, but the fighter is already the equal to a Rogue on 16 of 18 skills, so from a mathematical point of view that is not relevant. You could argue that in play the Fighter is going to use TM earlier on some other skill, so the Rogue will play better in his narrow focus area of 2 skills, but then you are saying the fighter is better at almost all the skills and all non-skill ability checks except those two specific skills the Rogue chooses to be an expert in.
Tactical mind is a great bonus. There is no disagreement here. But when you talk about limited resources it is the limited resource.
A person cannot claim spell slots are a limited resource and ignore that second wind is a more limited resource, and a person also cannot compare the number of uses of second wind to spell slots as if they are equal either.
At level 2-4 it is not a more limited resource than a 2nd level slot, it is a FAR more plentiful resource.
If you cast a spell and do not use a spell slot that spell slot then that resource is not used. Same with Tacitcal Mind, if you use Tactical Mind and do not turn a failure into a success then that resource is not used.
In most cases spell slots are used regardless of success or failure, it is not so with Tactical Mind. So when talking about how limited it is, you need to consider the chance the resource is used up. Withe a reasonable/normal DC it will take 4 skill checks to burn a single use of tactical mind with a 0 skill bonus. So 2 uses of tactical tactical mind goes A LOT further than 2 2nd level slots.
I never claimed rangers were the best. I did claim that they have spell support. An example of this in play is the jump spell. This is a good spell at those levels. One jump spell over three rounds is one spell slot used three times when the fighter in that range can use tactical mind on an athletics check twice and still not keep up, and then be out of uses while jump is maintained.
Jump is not typically used for skill checks in the games I have played, nor are there typically 3 athletics checks in 3 consecutive rounds unless you are trying to escape a grapple and failing.
Tactical mind only applies to a single roll.
Not true
It does change a failure to a success when it works but still isn't guaranteed to work, and it still will only apply to a single roll, and will still be very limited in the number of uses.
It always changes a failure to a success if you use the resource. So yes from a resource perspective each use of Tactical Mind is guaranteed to work.
If I have 2 uses of Tactical Mind, never take short rests use both of those resources, I will guaranteed turn 2 failures into successes.
This is why your discussion on resources is flawed. Tactical Mind does not make every check a success. Neither does enhance ability, guidance, expertise or anything else. But every resource used on tactical mind does make a check a success.
Cantrips and rituals don't use resource, and spell slots apply to more than single rolls.
As I stated above, Guidance comes closer than anything else in the game to matching tactical mind in terms of power on skill checks in level 1.
I think the concentration and limiting it to a single ability make it less flexible, but it is the only thing that is typically going to be in play nearly as often in tier 1.
The problem with enhance ability is that it's a second level spell slot for advantage when the help action can create advantage without resources anyway. The benefit of enhance ability is that it also applies to many rolls and doesn't require someone giving up their action for that advantage, and it can be applied to checks where the help action might not be able to. No matter how a person looks at it, an hour of advantage is more then the two uses of tactical mind.
No it is generally less most of the time I look at it. It can be more, but I think that is rare that you will get enough checks in an hour to overcome the advantage offered by Tactical Mind. Not impossible, not unheard of, but not common IMO.
Bards at those levels have the extra skill, expertise, and jack of all trades without the resource use.
They have one less skill they have base proficiency in and two more they have expertise in.
They do have a +1 across the board on the other skills, so they are +1 better in 15 skills, -1 worse in one skill and +4 better in 2 skills.
They also have the spells that may or may not include applicable cantrips or rituals that don't have that resource use, and they have access to spells that leverage multiple actions instead of single actions for more efficient resource use.
Sure they have spells, the ones that are effective on skill checks are far more costly in terms of resources.
I think I covered all the Cantrips and 1st and 2nd level spells in the game that can be used for skill checks above. I may have missed a few.
In addition to that, bards might not use bardic inspiration on themselves but they do give it to the person with the best bonus
You mean give it to the fighter? Yes, the Bard in the party makes the Fighter in the party even better at skill checks than he already is .... and if you want the most likelihood of passing checks at this level, that is who the inspiration should go to for most skill checks.
There are some cases where the fighter has a low ability and another PC has a high one and expertise and in those cases those other PCs are better, but you are talking about a couple skills out of 18 that someone else is better in with Bardic Inspiration.
and that's better than the fighter using tactical mind on themselves with a poor base bonus.
Better than what?
If you are using Standarad Array the Fighters numbers are pretty good in terms of skill checks, better than most classes and better generally than a standard array Rogue in my opinion. The 8 Charisma and 10 Wisdom really hurt a Standard Array Rogue when it comes to being a skill monkey.
If you are using point buy every PC gets the same number of points to spend, so it is all the same.
If you are rolling, well then luck is the deciding factor in the base number.
This is not an argument that the Rogue is better than a fighter, if anything it is an argument they are worse generally because their standard array is worse.
Fighters have a good bonus on a very limited number of uses, but bards have the strongest overall set of abilities at these levels.
Fighters are going to be better at skill checks generally than Bards and far more uses than Bards have spell slots.
Moving into tier two the increased bardic inspiration die and font of inspiration becomes a better resource than tactical mind regardless of the bard's other skill benefits or spell support.
Tier 2 is different because now Tactical Mind actually becomes a much more limited resource as there is something else effective to use it on (Tactical Shift) and unlike Tactical Mind, Tactical Shift does use a resource every time you use it.
While the Fighter still has TM available though, the best use of BI for skill checks is still usually going to be the fighter. In some cases Reliable Talent can make the Rogue better, but most of the time across most skills the fighter is who you are going to want to give Inspiration to in tier 2 if he still has Tactical Mind available.
Reliable talent doesn't increase the ceiling of the rolls, no, but bounded accuracy already does that so it's moot. There's no need for bigger rolls to increase the chance of success. Using the same assumptions when reliable talent comes online the fighter using tactical mind has a 90% chance on those 15 DC checks but a rogue with expertise has 100% chance on 20 DC checks.
100% chance on DC 20 on up to 4 out of 18 skills but only if he also has a +4 Wisdom, Intelligence or Charisma and expertise in the skills to match that 18 ability.
In the other 14 skills a Rogue has a lower chance of making a DC 20 than a fighter with the same ability score and tactical mind available and he has a lower chance of making a DC 15 check in at least 12 of those 14 skills (with the same abilities).
As I said clearly above, 2 more Expertises and Reliable Talent close the gap quite a bit.
There is a reason I am focused on level 2-4 and that reason is these are the levels where Fighters are way ahead. They are not way ahead after this. They are still good, and ahead of Rangers I think all the way to level 20, but they are not ahead of Rogues at the very top of tier 2, or most full casters for that matter.
