It will not hurt racial balance
I give everyone a free feat at level 1, and variant humans can get 2 feats instead of 1. Even then, my players did not go picking variant humans for their characters. I think it actually emphasizes the equivalence between v. humans and other races, because now a human can start with a +3 to one stat (two feats that grant +1 and a +1 from the racial bonus), whereas any other race with a +2 to any stat and a free feat can also achieve a +3.
This benefits you as the DM, too
Depending on how you build your NPCs (if they are allowed to have PC levels), and depending on how meticulous you are at encounter design/balance, allowing everyone an extra feat is actually very freeing for you, too. Now, you can create NPCs that are leveled like PCs, but with a lot more customization options that the players also have access to.
When I create encounters, I keep track of CR and other balance considerations, but I also try to establish a sense of "fairness." As the DM, I have infinitely many options available to me already. However, when I use only NPCs that have the same restrictions as the players, it establishes:
The player characters are normal people in the world, because there are others just like them, establishing a feeling of integration with the setting
The world is made of a predictable and knowable set of rules, rather than an ever-changing set of rules depending on my mood as the DM, which enhances player agency
A sense of fairness, because the DM is using opponents with abilities that the players can anticipate if they put effort into it
If you have this style as well, then a restriction on the players is a restriction on you. Freeing the players will also free you.