Sounds like 5E is not for you because the same arguments you're making apply to attack rolls and saving throws in this edition. Higher level characters are not wildly more competent (independent of build choices) than low level characters outside of HP, potential damage output, and spellcasting effects. If you want to be so good at skills you never or rarely fail, the Skilled feat, numerous ways to access Guidance, Rogue/Bard dips, etc. exist.
It's not entirely true. Take attacks for example. While a level x fighter with a Belt of Str and a level x wizard with a Belt of Str have the exact attack bonus with a pointy stick, the fighter (and pretty much any other class that's expected to poke people with pointy sticks) has a bunch of stuff (extra attacks and whatever) that make it vastly more effective at poking people with sticks than a wizard. Which is all good, because wizards aren't supposed to be good at poking stuff with a stick, it says so on the tin.
Now, in regard to skills, it doesn't really say so on the tin that apart from Rogues and Bards nobody is supposed to be good at using skills, but that's the feeling I got out of the system. I used to play a Noble Elf Wizard in a 5e campaign who had Persuasion and, by virtue of Position of Privilege was using it quite often. From level 1 to level 12 my bonus in Persuasion has increased from +3 (2 prof +1 Cha) to +5 (4 prof +1 Cha). Taking the Lucky feat at level 12 was a significantly larger boost to my ability to use Persuasion than the previous 12 levels of character development.
Unbounded skill check DCs give more options for setting difficulty, but run into the problem I previously stated that players won't even try to do things they aren't completely maxed out to do and/or skill checks are no longer a challenge at all for the right builds.
Meanwhile, bound DCs run into another problem: everybody (including henchmen and everybody else you can persuade/coerce to try) will attempt to do anything, regardless of their competence in the task at hand because if you throw enough dice at something, a natural 20 will come up sooner or later. Build yourself a focus group of 20 clueless averages (10 in all stats, no skill proficiencies) and they can statistically succeed at any Hard difficulty task you set them on.
Thought experiment: What would you think if skill check DCs were all lowered by 5 and resolved on a d10 but no other numbers changed?
It would probably be more to my liking. You no longer have Joe Average solving difficult problems he's clueless, and the actual bonus your character provides will matter more.
EDIT: on the other hand, upon thinking about it more, that would also be a pretty big boost to rogues/bards, but I don't think it would be a problem.