I've both run and played in games that gave a free feat to all PCs at first level. It hasn't been a problem so far. Yes, characters are a little more powerful, but not hugely so.
Variant Human in my game loses the bonus feat and instead gets +2 to one score, +1 to another, 1 free save proficiency, 2 skills, 1 language. Out of 6 players, we have 3 humans, 2 wood elves and a dwarf. (In our other game where the DM left Variant Human as-is, we have 5 humans and 2 half-elves. So I figure my changes didn't make human too attractive. Our group just tends to gravitate towards being human heavy.)