I deeply dislike ability scores. While pretending to enhance your character concept, instead they straitjacket it, and they add a lot of unnecessary math and introduce "trap options" for inexperienced players. They were a bad idea in 1E and they are a bad idea now, and the only reason they persist is their extreme sacred cowness.
I have taken a stab at eliminating ability scores from D&D. There is no systemic obstacle to it--nothing in the core game engine really requires ability scores. The problem is all the zillions of references to ability scores that have to be examined and removed. I concluded it wasn't worth the effort for me personally, but I'd be very interested if some enterprising third-party publisher made a 5E clone with ability scores excised.
The essence of my approach was to fold ability scores into proficiency. Instead of starting at +2 and going to +6, your proficiency bonus starts at +5 and goes to +9. When you would get an ASI, you can instead choose between a feat and +1 to your proficiency bonus; you can't increase your bonus more than twice this way. (This mimics the progression of starting with a 16 in your primary stat and increasing it up to 20.)
With this system, there are two ways to roll a d20: With proficiency or without. When you make an attack roll, if you are proficient with the weapon, you roll 1d20 + proficiency; otherwise it's a straight 1d20 roll. Likewise for skills. For saving throws, I would be inclined to go back to Fort/Ref/Will, which is simpler and avoids the nuisance of "major saves and minor saves."
But then there are all the little nuisances:
- You need a weapon damage modifier to replace the stat bonus. This might come from your class, perhaps. Or it could be built into the base damage of the weapon.
- All classes will probably need a hit die boost, since they no longer have access to bonus hit points from Con.
- Armor is currently designed so that high-Dex characters favor light armor and high-Str characters favor heavy armor. Obviously this does not work if Dex and Str don't exist, so you'd need another way to calculate AC.
- Expertise is now way better. The easy solution is to cut it in half: Expertise adds 50% of your proficiency bonus instead of 100%.
- What do you do with feats that grant +1 to an ability score?
- All the gazillions of class and subclass abilities that refer to ability scores have to be replaced.
- Calculating encumbrance.
- Etc.