If classes are going to grant abilities just about every level, then I think race should do something similar as well. I'd advocate for 1st/3rd/5th/7th/9th (or 1st/5th/9th) abilities myself. And preferably not "+1 to X stat".
1E and 2E didn't really need this sort of thing as there were no decision points during leveling, and other than spellcasters very few classes gained new abilities at level up - beyond better THAC0, hp & saves.
One of the things I've done with races in my 5E Frankenstein is:
Dwarves: If you choose a class/subclass that doesn't cast spells, you get Magic Resistance
Halflings: If you choose to take a -1 to Strength, you get a free Dex-based skill proficiency and PB advantage on Dex skill checks per day.
Elves: A feat that allows you to use Trance to change one skill Proficiency during a long rest.
And so on.