The variant I rolled up was 3e style gestalt. Like, species was a class.I had thoughts along these lines a while back, but I went in a slightly different direction: the base idea was "what if your race was worth a whole class level?" which led me to relate each race to one or more classes:
Dwarves were initially fighter-types (more focus on defense), though barbarian wild dwarves and monkish dwarves also showed up. Elves were half- casters but what kind of magic was set by subclass (wood elf = druid, high elf = wizard, blood elf = warlock, etc.) Halflings were roguish, gnomes were either artificers or something more fey-like, dragonborn were sorcerers and/or beast barbarians (but dragony), and planestouched were warlocks.
For humans, I ended up making them the cleric race, both because they needed a niche beyond "versatile", and because I hadn't found a cleric-y race yet. I liked the answer as it made humans a little more distinct while keeping them versatile (clerics can be a lot of things). It had some other upsides (ie justifying the human pantheon while more other races were monolatrist)
I never got around to writing out stat blocks (I was more interested in the worldbuilding implications) but it's an idea you could borrow.
Elves would learn elvish magic along side whatever else they adventured as.
I then over complicated it. You had background/species, subclass and class.
Every 4 levels you got 4 class levels, 2 subclass levels, and 1 background level, 1 species level. Humans might get as their racial "2nd background" or whatever.
Multiclass PCs would swap their subclass for a 2nd class.
The problem was I find you want to establish identity for species/background/subclass early, which this mechanism didn't provide.
But the idea that high elves are all wizard hybrids was pretty neat. (and wood elves druid hybrids). The other class could be tweaked to be a roughly equivalent class but not identical (so the elf class isn't exactly the same as a wizard, just close).
High Elf: Wizard
Wood Elf: Druid
Halfling: Rogue
Dwarf: Fighter
Orc: Barbarian
Gnome: Artificer
It would generate a really strong species identity.