First, I'll just say I really don't like skill granularity. By which I mean, I don't like 'Jump' or 'Climb' or 'Arcana' as skills, because it limits what skills you have to the list available, and if I wanted 'Climatology' or 'Electrical Systems', it means I'm opening up the skill system to either so many choices (with fewer selections) that it becomes analysis paralysis, or at best FOMO for not picking 'the right skills' for the campaign. As such, I far more prefer attribute based checks from the DMG.
However, to address your concern in your opening paragraph, I think it requires a paradigm shift in what 'success' looks like for each skill. For instance, I'd downgrade a jump success from 'you make it over the ravine and go along your merry way' to 'you jump over the ravine, but end up prone.' A success with drawback would be 'you barely make it across, finding a shrub or tree to grab onto. It's going to take you a round to scramble up over the edge of the ravine.' A crit success becomes the new normal 'you make it over the ravine and go along your merry way'.
Stealth and any other opposed check is actually pretty easy, imo - drawback: opposed has advantage; success: opposed is normal; critical success: opposed has disadvantage. That's the mechanical side though. Describing the why of those three results might be a bit harder on the DM though...
1) I like that, if you're using the skill system as presented, there should be a mechanical advantage (not Advantage) to being trained that's more than just 'you have a slightly better chance at success than some uneducated bumpkin.'Some things I've toyed around with (but never play tested)
1) Untrained skills have a DC cap of 8 + proficiency bonus. DC's above that cannot be attempted by someone without proficiency in that skill
2) Expertise grants a "take 10" feature; instead of rolling, the character may count the check like the rolled a 10. This cannot be used in combat or other hostile situation
3) Not really a mechanical change but a table of DC's to use as a reference is really needed. My general opinion is skills tend to be too high of a DC; someone who is trained in something should probably be succeeding at that thing like 75% of the time. The norm DC should be in the 10-12 range; higher is notably hard things to do
4) Yeah, something like total fail, partial fail, success, and total success would be nice (but see above comments)
2) As an option, or replacing the double proficiency bonus? I like options... I'm not sure I'd like it completely swapped.
3) One reason I really like the flat DC 15 for everything. Removes the need for tables of DCs and replacing it with a smaller list of drawbacks and crits ;)
4) see 3 ;)
I think it also depends on how the table runs skills. If the DM is running the story and then comes to a decision point and states "you've come to a walled dead end, there's a gap at the top the party can squeeze through, roll Strength (Climb)" it definitely takes away agency, as the players more than likely won't question it, and just roll climb checks and continue on. If instead, the DM says "you've come to a walled dead end, what do you do?" Then the players might start rolling perception checks and climb checks or cast spells... less is definitely more when it comes to options and agency. I don't recall if there's anything explained that way in the DMG. I'm sure Psyren could point to the exact page number, if it exists. But it's a skill (no pun) that DMs learn over time, if they didn't learn it while playing at the feet of good master.But - none of this addresses what might be my core problem. There's no agency in skill checks. It's just use the obviously best, highest check you possibly can. I guess something like the Varying Success system could address some of this? Have poorly matched skills have terrible downside risk? But again, that's just a lot of DM time to make it feel right.
100%, it's why I'm moving away from skill checks to just blanket ability checks. "Can I make an Intelligence check to figure out how this puzzle was put together" is more engrossing than "I make an Intelligence (Investigation) check. What do I find?"
The only pushback I have on this suggestion is that the community (at least in my neck of the woods) that engage in anything other than D&D is trifling small... I think it's why there's such a vibrant D&D modding community - when everyone knows the base rules of the game, it's easier to bring in ideas from other games, instead of trying to change what game people are actually playing. I wish there was a more robust gaming culture I could connect with... but it's really 5E + homebrew, or nothing.