First off, my comments are from the viewpoint of no feats, as I don't allow them in my game.
The problem with the old school way of improving characters is that every character improved the same, based solely on level. An 11th level fighter had particular THACO and saves, which were the same as every other 11th level fighter. I remember one reason my group stopped playing 1E was that characters tended to be the same - most mechanical customization depended on which items a character possessed.
If we view ASI's not as actual physical/mental changes in a character, but instead as the result of training and practice, ASI's give a player considerable flexibility in customizing the character. A Dex-based fighter will look and fight differently from a Str-based fighter. A bard with high Cha, but low Con will be different from one with moderate scores in both abilities.
One could create feats that effectively mimic ASI's. For instance, a feat that improves a character's bonus to initiative, Dex skills and saves, while also improving their ability with ranged weapons (bonus to attack and damage). But it's just so much easier to base all those things on the stat and improve the stat. The one problem is that characters begin to look the same, as stats have so much impact on class abilities, but I'm unsure how to avoid that while retaining bounded accuracy.