That would bring numbers up higher than I like to see at level 1, but if it works for your table--great!
The entire concept of moving ASI's because they cause players to only pick certain races because that gives them an extra 5% to their main stat seems odd to me. I do not get it. It is changing a pillar of D&D for little or no reason imho. So when it was expressed that ASI's should move, but racial traits can stay because they don't "favor" a class. I pointed out that many racial traits do favor classes, melee, spellcasters, bards, and barbarians. If you refute that these things do not favor a class, great. I can't argue that. And in fact, like to encourage that type of character construction. For example, a barbarian that hides all the time. Obviously dark vision would be beneficial for this, as one cannot hide in the dark holding a lantern or torch. One can avoid tripping on noisy things if they can see where they are stepping. But the metagamer in me knows that it benefits the rogue more because they are the ones doing most of the stealth work. So yes, I like your thinking. And if more people approached it the way you just did, then we wouldn't need to have this discussion about moving ASI's. As any bonus anywhere can help any class. My tiefling barbarian can finally use intimidation because they have a charisma bonus. They may not have optimal strength, but they have intimidation, as an example.
Well, I don't know... I felt like I was going in circles, so here is the short(er)-version:
My choice would be remove ASIs from race completely. No floating, no moving to class/background--just get rid of them. Why? Because if you want a high score someplace because you feel it either represents your race, class, and/or background--just put a high score there! You want it to be higher? Fine. Roll and hope you have good luck or bump it during game play, that's why 5E offers ASIs as you level (they aren't for feats--those are optional
).
Once they are removed, replace them with racial traits (if needed) to show that a particular race is good at whatever. Yes, some of those "goodies" will still lend themselves towards certain classes--but really the vast majority of them can benefit any class
if you play to them! That's important. If you can't think of a way (as I described before) then odds are you will feel at times anyway the PC is sub-optimal.
Finally, cap scores so the ones who show reason to have superior scores can have them. Maybe that is 18 and 20 for caps? Will that +1 make a difference? Depends on how important it is to you. The ASIs hardly matter to me. I played a Goliath Druid and used the STR +2 to get him from 8 to 10. Yeah! I didn't have to spend any points on it and it allowed me to put those points from point-buy where they
would matter to me.
Anyway, since most people don't like lower numbers (I feel
so alone at times! (j/k)), the next best option is just leave them alone. If you float them, they are meaningless. If you move them, the meaning shifts from race to class and/or background. If you just bump methods, you are basically simply saying "I want higher scores, I don't care how."
Failing all that, just keep them as is. As I showed with the Goliath Druid, I was able to put a lower score in STR because I got the +2 ASI for race, and that allowed me to have a better score where it mattered. If I had played a race with a bonus to WIS, I would have had to spend points to bump up my STR from 8 to 10 anyway...
Ok, I need a break... Time for lunch.
-------- The rest was the start of my original post, read at your own risk.
---------
Well, just so you know my position, IME most players pick a class first and then look over the races that they think will fit the class concept. Because ASIs get associated with where they want to be strong, many players pick the races that compliment the key abilities for their chosen class. As others have pointed out that 5% might not seem like a lot, but depending on how you view it--to them it is. Human nature in general is to want the best--and in D&D that includes ability scores. Players will more often complain due to high scores. Admittedly, I am different. I have played for nearly 40 years and prefer when my character fails more often than succeeds--it makes the game more challenging for me.
As for the other racial traits favoring certain classes? Certainly, as I expressed I
agree with you on that, but maybe just not to the same degree. A prime example to me would be the Goliath with Powerful Build. Such a PC can wear heavy armor, tote around a dozen of weapons, etc. more easily than even an Orc (which also is +2 STR) can do. This is more of an issue at our table because we use the variant rule for STR and encumbrance. So, a STR 16 can carry 80 lbs before suffering penalties. That is basically just plate armor, shield, and some weapons. You have hardly no room for gear as well. But, with Powerful Build that double to 160! Now, pack everything on you want for the most part! Classes such as rogues and wizards, etc. who normally don't carry much, won't benefit as much from that racial trait, they might not effectively benefit at all TBH.
So, for me, moving ASIs are more for those players who value that 5%, which IME is most of them. You will rarely find a PC of mine with an 18 until tier 3 and I've only ever had
one PC with a 20 in
anything! Occasionally, sure, but it is not often at all. I would rather those racial "highlights" (?) be shown by other features. Does being part of a "stronger" race mean more of them will likely be warriors? Probably. People naturally gravitate towards their strengths.
What it comes down to often is the player mindset and understanding in 5E you don't
need to be +5 or +6 at level 1 to be effective and have fun with that PC. +4 works great IMO and I am okay with even a +3. Sure, I don't want to
completely suck and have a +2, but it can still be done.
Now, you bring up an interesting example. The tiefling barbarian. (Thank you, by the way!) Yes, you can use your CHA +2 and be good at intimidation. But where are your other scores? Honestly? Odds are you will have a decent STR and DEX and CON. Why? Well, because of the Rage feature. The damage boost is only there with STR-based weapon attacks. You can't gain it when using DEX. DEX is still an uber-stat, important for just about so much you can't avoid it, especially since you can't wear heavy armor (you aren't proficient...). And CON? Well, that is pretty self-explanatory IMO. So, that leaves INT, WIS, and CHA. Tieflings get INT +1 and CHA +2. Well, if you have already gotten decent scores in STR/DEX/CON, odds are your mental scores won't be that great without having a severe problem elsewhere. By the time you put your CHA +2 into CHA, you might have a 14 in all likelihood? Well, that total +2 modifier will probably be as good as your STR anyway, which a lot of groups use for "physical" intimidation rolls.
