And (as I have had to say far too many times), if it's a resource everyone gets, it doesn't count. Everyone gets a background and four skills and racial/species features. In "One D&D," everyone gets a background feat. If it's something legitimately everyone gets, it does not count as an advantage addressing the gap in what the Fighter itself brings to the table.
This is a trap and a mistake!
If you add something that both a wizard and a fighter get, and using that feature is an alternative to (say) casting a spell, this can make fighters better than they where but have almost no impact on wizard capabilities!
In short, in a game where some PCs have "I can change reality with magic", making skills better
for everyone actually closes the utility gap. Even if wizards can also use skills.
The size of the gap
matters. Imagine a game where everyone had near-wizard spellcasting, near-rogue skills, etc. And the wizard got slightly better spellcasting, the rogue got slightly better skills, and the fighter got 4 attacks per turn. The lower gap in areas
besides attacks per turn reduces (and maybe reverses) the gap between fighters and other classes.
I am belabouring this point because if you hold it as an axiom, you miss entire ways to solve design problems and you discard viable fixes as irrelevant.
So yes, background skills actually boost the fighter more than they boost the rogue. Despite the fact the rogue has access to the skill as well. Having more baseline options to interact with the world boosts non-spellcasters more than it does spellcasters. Despite the fact that they are available to spellcasters as well.
In both cases, the diminishing marginal returns kicks in, and reduces the benefit to the "richer" party.
And if enough of it is done, the fighter doesn't have to be
that much better at combat to make up for being poorer in other areas.