Xeviat
Dungeon Mistress, she/her
Secondly, a fighter should really be the most common and relatable of the classes amongst common people. Someone who adventures by the strength of their arm should be more understandable and less "weird" than someone who casts magic because they have dragon blood, or even someone who casts magic out of their lute.
I've brought this up a few times before, and definitely once here. I firmly believe the Fighter should be the one who is most relatable to the masses. This is a hard thing to reflect mechanically, though, but in the world fighters tend to be the leaders. In archetypical party building, the fighter easily fits into the role of the "hero". The other classes all have "weirdness" to them, or the perceived dishonesty of the rogue (rogues fight with trickery, not stand up martial talent).
Giving the fighter an extra skill wouldn't hurt; I like the notion that they should have more trained skills than the wizard, your point is solid. I'm not sure I'd expand their skill list, though; as other have pointed out, backgrounds can provide other skills that are needed beyond the base fighter stuff.
I do think the fighter needs a 1st level class ability that speaks to them, and then their subclasses need some more identity. As much as I want to rearrange things so all subclasses are at 1st level, it might make the most sense for it to remain at 3rd for the Fighter, unless one were to make a more generic fighter subclass.