Well that subclasses. Subclasses muddy the waters. Battlemaster and Arcane Archer both let the fighter outburst the Ranger with any other its subclasses.
This is a problem.
Counter-point. No it isn't. It has literally never once been mentioned as a problem anywhere that I have seen. And it isn't just a subclass thing. Again, I mentioned Action Surge. That is a burst move. And they can do it at range.
You did the math on the Fighter not the Paladin.
If you are saying the 5e ranger has no niche, what niche should the Ranger have in your opinion and is it good at it?
Um... no? I did the math for the paladin at the very beginning.
An archer Paladin before level 11 is going to be dealing 2d8+modx2, with the option to burst an extra 2d8 to 4d8 damage on a hit.
An archer Ranger in the same level range is going to be dealing 3d8+modx2 with the option to make that 3d8+2d6+modx2 for every turn. A paladin would have to be burning a LOT of smites to make their attacks more damaging, more consistently.
Now, post level 11, it changes up, because the ranger doesn't change, but the paladin ends up dealing 4d8+modx2 with the option to burst, but the ranger needs something at level 11 as well. So, I don't think we are really looking at paladins potentially out-damaging rangers at range. Sure, they have slightly better defense, but it ends up being a single point between half-plate and full plate. The bigger difference is usually due to having a shield.
And this doesn't account for the melee ranger. Again, if we are talking sustained damage potential, the melee ranger who decides to go PAM (because they can) does 2d10+1d8+1d4+3d6+modx3. Which is REALLY impressive damage that the Paladin only really starts beating post level 11.
I even did it twice
Ranger, hunter's mark, level 6: 3d8+2d6+8 = 28.5 x 0.7 = 19.95. Repeats every turn
Paladin, Divine Smite, level 6: 5d8+8 = 30.5 x 0.7 = 21.35... uses their most powerful spell slot.
This is barely a single point of difference in the Paladin's favor. And they can only do this TWICE per day. Meanwhile the ranger can do this for 4 hours? And that is just burning 1st level slots. Is this really cutting the ranger out of their niche? To lose to the paladin on two attacks out of the day? Sure, it increases the more the paladin smites, but we are talking this only being an issue for 2nd level, 3rd level and up smites. Before we reach level 11, where I admit there is a problem.
And, no, I'm not saying the Ranger doesn't have a niche. I'm saying Ranged Burst damage isn't their niche.
The problem is the Fighter, Barbarian, and Paladin all out damage the Ranger AND out Tank it AND matching it in skills is just a feat away.
The OD&D ranger is still Bad. You did the math.
So you can't really buff or expand the Paladin, Barbarian, and Fighter until you determine what the Ranger is good at.
Balance. Some game Balance, please.
I literally have two posts above you showing that the Fighter, Barbarian and Paladin DON'T outdamage the ranger. But okay, let's just get comprehensive with this. I'm not going to bother with accuracy numbers. Let's just throw this down. All mods will be +4, just to make my life and the math easier. All level 5-ish. I'm also not going to use Weapon Masteries, all of them should get them, so it would all be identical.
Sword and Board
Hunter Ranger: 3d8+2d6+8= 28.5, used a single non-concentration 1st level spell. Can repeat every turn*
(Theoritical) Battlemaster Fighter: 4d8+8= 26, used half of short rest dice, can only do so once more.
Berserker Barbarian: 2d8+2d6+4+8= 28 used a single rage (1/3 of total daily) can repeat every turn*.
Paladin (subclass shouldn't matter here, and I don't have a good example): 2d8+3d8+8 = 30.5, used a second level slot. Can only do this once more. Downgrades to 26 damage with the four 1st level slots.
Ranking: Paladin (for two attacks per day), Ranger, Barbarian, Fighter
Polearm Master
Hunter Ranger: 2d10+1d8+1d4+3d6+12= 40.5 used a single non-concentration 1st level spell. Can repeat every turn*
(Theoritical) Battlemaster Fighter: 2d10+1d4+3d8+12 = 39 used 3/4's of short rest dice, cannot be repeated without rest
Berserker Barbarian: 2d10+1d4+2d6+6+12= 38.5 used a single rage (1/3 of total daily) can repeat every turn*.
Paladin: 2d10+1d4+3d8+12 = 39 used a second level slot. Can only do this once more. Downgrades to 34.5 damage with the four 1st level slots.
Ranking: Ranger, Paladin (for two attacks per day)/Fighter (once per rest), Barbarian
Longbow
Hunter Ranger: 3d8+2d6+8 = 28.5, used a single non-concentration 1st level spell. Can repeat every turn*
(Theoritical) Battlemaster Fighter: 4d8+8 = 26, used half of short rest dice, can only do so once more.
Berserker Barbarian: 2d8+8= 17, cannot be improved with class abilities. If using a thrown weapon, like a handaxe, then we can get 2d6+2d6+4+8= 26 with the use of Rage (1/3 of daily total) and repeat every turn*
Paladin: 2d8+3d8+8 = 30.5, used a second level slot. Can only do this once more. Downgrades to 26 damage with the four 1st level slots.
Ranking: Paladin (for two attacks per day), Ranger, Fighter/Barbarian (handaxes)
Dual-Wielding
Hunter Ranger: 3d6+1d8+3d6+12 = 37.5 used a single non-concentration 1st level spell. Can repeat every turn*
(Theoritical) Battlemaster Fighter: 3d6+3d8+12 = 36 used 3/4's of short rest dice, cannot be repeated without rest
Berserker Barbarian: 3d6+2d6+6+12 = 35.5 used a single rage (1/3 of total daily) can repeat every turn*.
Paladin: 3d6+3d8+12= 36 used a second level slot. Can only do this once more. Downgrades to 31.5 damage with the four 1st level slots.
Ranking: Ranger, Paladin (for two attacks per day)/Fighter (once per rest), Barbarian
* Ranger has a 1 hour, nonconcentration limit. Barbarian has 10 minutes with bonus actions
So, there we go. Now, are there some assumptions in all of this? Yes. There are also ways for the Ranger damage to IMPROVE, which would be much harder for the other classes.
Now, again, I will admit. This changes at level 11. I know that. You know that. We all know that level 11 is poorly designed for the Ranger and the poor barbarian. But you want to know what the ranger is good at? Consistent, turn by turn, damage. Look at the rankings. They are either in second place, behind the paladin using their biggest spell slot (which they can only do twice), or in first place with the paladin in the same boat and the fighter burning SIGNIFICANT resources.
But, for balance, game balance people, the ranger is slightly less tanky and naturally better at skills.... which seems like the fighter and barbairna need a buff. The paladin is fine, because they got some non-damage buffs that are very good.