I'd apply the same reqs to single classing, actually.One thing no one else has mentioned is that ability scores are used for multi-classing viability.
With Shadow STR drain, the PC can regain STR after short or long rest. Anyone know what is RAW for regaining INT after an Intellect Devourer reduces a PC to 0 INT?
The big, main factor is the random distribution of 3D6 (4D6 drop lowest is a heroic 3-18 curve) creates a great curve for random attributes: and random rolling in the traditional style is the assumed way to play.
Alternate HPs can mean at least a couple different things. Firstly, it can mean taking ability score damage instead of HP damage. Secondly, it can also refer to alternate methods of calculating HP totals.
I know it's a houserule, but I allow PCs to automatically succeed on un-opposed ability checks where their score is equal to or greater than 5 + the DC of the check. So, if you had a 15 Strength you'd never have to make a Strength check to force open a door (for example) with a DC of 10 or less.
One thing no one else has mentioned is that ability scores are used for multi-classing viability.
Very little. It's one of the failings of 5e. In older dnd, attribute checks were made against your score, eg want to break down a door, roll d20 and get your str score or less. ... The roll equal or under approach also allows a lot of other cool things.
Well first Thing they do for you is to convert a value between 8 and 20 (which you can optionally estimate with dice and racial modifier) into a -1 to +5 Bonus.
I also houserule in my game that casters Need 10 + spell Level in their casting Attribute to cast a spell, (like in former 3e times e.g. 15 is needed to cast 5th Level spell)
Another houserule i use is Str Minimums for armor (full plate requires 18 STR in my games)
So maybe the scores are in not much use RAW but that doesn't prevent you from utilizing them.
With Shadow STR drain, the PC can regain STR after short or long rest. Anyone know what is RAW for regaining INT after an Intellect Devourer reduces a PC to 0 INT? Or would this be more of a DM ruling kind of thing? If it is a ruling, I’d be inclined to allow full heal on Greater Restoration and some kind of slow heal on rests (1 point per short rest, 2-4 pts per long rest).
Additionally, there are some magic items which enhance abilities. Some of the Ioun Stones come to mind. Some spells might do the same
Good call. Enhance ability is the obvious one that comes to mind, but I checked the description and it actually doesn't raise your scores, it just gives you advantage on checks (and a few other benefits specific to different scores, such as temporary hit points). Does anyone know any examples of 5e spells (or items besides Ioun stones) that raise your ability scores?

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.