This wasn't so much a problem as I just wanted to work it out, but what if you gained ability scores from your class AND your race? This is a trial by fire version and I don't have everything worked out, but here goes nothing:

RACIAL ability scores grant a 1 from your highest bonus if there is no subrace ( 1 STR for Half Orcs, 1 Cha for Half Elves, 1 Cha for tieflings), from your subrace if you have one ( 1 Str for Mountain Dwarves, 1 Int for High Elves, 1 Con for Rock Gnomes).

Humans are a special case, gaining a 1 to 2 different ability scores, as well as two choices of either a language, a tool or a skill. So skill language, tool skill or tool language.

CLASSES give a 1 to each of two scores at level 1. Multiclassing does not grant these bonuses. This one was a bit challenging, and casters present a unique problem, in thst they can be built single stat. All in all, these scores try to reflect the class they are a part of. You cannot gain more than a 2 bonus to any one ability score.

Barbarian: 1 Str, 1 Con

Bard: 1 Cha, 1 Dex

Cleric: 1 Wis, 1 Cha

Druid: 1 Wis, 1 Int

Fighter: 1 Str or Dex, 1 other

Monk: 1 Dex and Wis

Paladin: 1 Str and Cha

Ranger: 1 Dex and Wis

Rogue: 1 Dex, 1 Int

Sorcerer: 1 Cha, 1 Con

Warlock: 1 Cha, 1 Str or Dex

Wizard: 1 Int, 1 Wis

Feel free to help with the scores, I'm trying to keep them all viable.

This removes sub optimal builds in a way, as well as showcases the work a particular character put into becoming who they are.

Edit: Made a slight edit regarding multiclassing, I'll get the saving throws for single stat classes when I'm at a phb.

Edit 2: Got the scores worked out a bit better, with the exception of fighter, all classes with a single multiclassing prerequsite gain a 1 to each of their saving throws. All classes with two multiclassing prerequsites gain a 1 to each of the prerequsites, and fighters gain a 1 to Str or Dex and a 1 to another score, and put a hard cap on the stat bonuses.