That is a completely separate discussion. Personally, I think the differences should be on the minor side and mostly flavor, because i care more about giving players freedom than making sure x race is better in y class.
Maybe give each race a selection of feats they can choose at lvl 1? Some overlapping with other races, like toughness, some unique to them. Would certainly help in getting away from race x->class y thinking.
It's not completely separate. ASIs relate to the default characteristics of races in one's game-world, which affects how NPCs act and how players relate to the different races. Are elves more naturally nimble but less hardy and strong than humans? Such a distinction might mean that elves, as a whole, are very wary about entering a human's (or an orc's) melee range because they know they could be easily hurt. It also allows a player to play against their race: putting most of my elf's stats into strength or con (or another example: being a very smart orc).
Without ASIs and/or sufficiently distinct racial abilities, you lose some of this characterization. The DM can of course assert that e.g., orcs in their world are dumb and gnomes are smart, but I want officially published D&D books to have racial distinctions by default. A DM can then optionally remove and/or change those racial traits.
Racial feats could be fine in addition, as long as they are distinct and not just "you get access to this spell." But I do think that there should be a certain base level of ~unique differences in D&D races.
