So, I would be okay with the proficiency bonus going up by one at every level, actually ("bounded accuracy" was never anything I felt like I really needed but I won't deny that broadly speaking it seems to be working). But having characters well, be proficient in things that they aren't proficient in, for lack of a more precise wording, I'm not sure why anyone would want that. It would leave ability scores as the
sole point of differentiation between characters of different classes.
Surely the level 20 wizard is a bit better with a longsword than the level 1 wizard,
Why? This doesn't seem like a "surely" to me, it seems like a dubious proposition at best. The wizard has concentrated his training on, well, wizardry. Why would he waste any time learning how to swing a sword considering that he can reshape the very fabric of reality through will alone? I just don't see the basis for this assumption.
The level 1 fighter with 10 wisdom and no perception proficiency is typically as good as the level 20 fighter at that skill. Why? Does it really make sense for that to be the case?
It almost sounds like you complaining about the presence of meaningful choices during a character's development. In between being Level 1 and being Level 20, that fighter has multiple ways to improve their perception bonus, including several ability score increases that could be applied to Wisdom, certain feats, and so on. When the fighter chooses to increase Strength by 2 so that he can swing his longsword better and do more damage with it, rather than increasing his Wisdom by 2 so he's more perceptive, that is one of several meaningful choices that a player makes for their character between Level 1 and Level 20.
It does make sense for me that a 20th level fighter that has not chosen to invest at all in their perception would be less perceptive than a 10th level fighter that had chosen to invest in being perceptive.
I can't really see any implementation of what you're asking for that wouldn't make all characters feel much too samey.