I want to get rid of the tiny modifier naughty word thinking around skills and add a more robust system that gives you some things that you can do, something that actually improves as you level. Less fiddly bits and pointless numbers. Less GM fiat. More real power.
I understand conceptually what it is you are trying to accomplish with your system. And yes, for some tables it might be useful. Specifically tables where players do not know DMs and how they will act and react to player requests. So if a game had the add-ons you propose, it would be fine. But I just don't think it is necessary.
Reason being... the answers you are trying to get permanent answers for can just be gained using context clues-- both in-world / in-game, as well as out-of-game meta knowledge of the party, the table, and the sorts of narrative hurdles the DM puts in front of the players in an effort to make the story interesting.
Your barbarian scenario for instance-- can they bend the bars to get through a gate? Yes, we could have your system wherein it is stated on a chart in the book "at this character level, at this skill level, and with DCs between X and Y... success is automatic." That is doable. Some games might even have that for all I know.
Or we can have the current situation-- the one with 'DM fiat' that you apparently do not care for-- where the DM chooses a DC based upon the knowledge of the the thing being tried, the character, their level, their skills, how important to the story is getting stuck behind this gate? etc. And that DC might as well be automatic if they make the DC low enough. Or the DM might (using yes, DM fiat) decide that a check isn't necessary at all (again, based upon character level, skill level, and whether being unable to move past the gate is at all important to the story).
Players will learn through play experience when things become automatic successes. Because DMs will stop asking them for skill checks altogether when doing those things. At some point when your acrobatic character says they want to scale up the side of the building onto the roof and the DM knows you have a +12 to your check and Advantage on those types of checks... they just won't bother asking. You just can do it. And that to me is a learned skill for players AND DMs that I personally believe is more worthwhile then just checking a table in the book that says "You can do X automatically". It's learning the entire process of skill use and DC creation and both players and DMs learning how to be reasonable with making ideas and suggestions and hurdles and narration based upon the checks. To me, that's learning how to play the game of Dungeons & Dragons... not just knowing everything as a player you can and can't do by checking a chart. You wouldn't even need a DM to play the game if that was the case.
But... that's just my opinion, I could be wrong.