The "race" will split up between "species" and "culture".
Backgrounds can handle the culture.
Eh... Dunno. Culture and Background feel different, to me.
For example:
Species: Elf
Cultures: Wood elf, High elf, Dark elf, Sea elf, etc. Each of those has a communal cultural view of how things get done, how people work together, how they view outsiders, what they feel
comfortable with.
Backgrounds: Maybe you spent a few years as a conscripted soldier, or were apprenticed to a blacksmith, or ran away from home to join the pirates on the high seas.
So Species is the intrinsic attributes of a race: some common stat increase, size, speed, darkvision, and expected lifespan. Dragon Breath for Dragonborn. Fey nature for elves. Etc.
Culture (in a generic, tropey sense) feels like the differences you might expect in a mercantile culture, vs a warlike culture, vs a diplomatic culture, etc. It affects the default languages, and probably a default skillset. You gain the benefits that everyone living in this culture is likely to have.
Some cultures might use direct physical conflict as a way of settling disputes. Some might use formal duels. Some use a strict legal system. Others might run on bets and wagers. Some are warlike. Some are reclusive. Some want to strike a deal. These are all broad, shaping influences on your character, and generally connect to some skill (athletics, gambling, dueling weapons, law, engineering, etc).
So, perhaps an orc tribe with a warlike culture will have everyone proficient in athletics, because that's just how differences are resolved. You might end up as a wizard, but your cultural upbringing influences how you approach solving personal issues. Culture is broad societal influences.
Background, on the other hand, is more about your own personal upbringing. Not everyone in a mercantile culture is a merchant. You might have been a caravan guard, or a sailor, or a blacksmith's apprentice, or a street urchin. Background is personal skill development — often tool, weapon, and armor use, with perhaps some supporting skills or languages, along with some ability that's useful enough to not be completely ignored the entire campaign.
I could see Species and Background giving you stat increases, but not Culture, which is more about skills.
"Elven woodland soldier" and "dwarven magi-tech miner", make appropriate backgrounds to choose from.
That unfolds to way, way too many backgrounds very quickly.