Also, you left out something crucial in your second bit there.
What is the Fighter contributing that literally any other character could not? Because that's what actually matters here. 5e skills are weak (particularly in comparison to 4e, where skills were explicitly presented as extremely broad and powerful), and even if they weren't, Fighters don't get anything to help with that. Literally all characters get at least 4 starting skills, two of which don't even come from their class. Unless the class provides something more than just having skills, it's not being a Fighter that let you do it. Period.
Everyone else gets things they can contribute specifically because they are the class they are. Fighters do not. That sucks. It should not be.