I guess I look at it like this. 5e uses a background mechanic to largely emulate skills. It also has feats that are broad and more general.
Re: backgrounds. I think this is where the biggest bonuses to skills checks should come from, and not the class itself. For example, I grew up on a farm. But that's not my job (class). My job is in tech. But I guarantee you that my farming skills are better than someone who didn't grow up on a farm but went to AG school. Experience always trumps education. So in that context, if my background was something like a hunter even if my class is a cleric or something, I should have outdoor tracking at least as good if not better than a level 1 ranger who's background was street urchin or sage or something. Currently, that's not the case.
Re: feats. I would rather see an expansion of feats that replicate class abilities, and then go to what I listed above; few classes with several subclasses. For example, a barbarian is just a fighter subclass with rage and unarmored defense. From there he or she could grab the berserker feat which would grant some of those barbarian abilities. Or grab a totem feat to grant those abilities. AFter all, that totem stuff a barbarian gets is a cultural identity, so why do only barbarian classes get it? Meaning, a shaman from the same tribe, or a scout wouldn't be able to have the same cultural identity? Doesn't make sense to me.
I get why earlier editions had tons of classes, but 5e is built with feats and backgrounds which makes it totally possible to do what I'd like to see them do.