It seems to me that the issue stems mostly around the concept of 'can do/can't do' being resolved via RNG in an unsatisfying way. Because there is a 1-20 range (i.e. 19) in success/failure, it seems that for something to be easy for one character it has to be a coin flip for another (at worst) and likewise, something which is impossible for one character is a coin-flip for the pro. I'm assuming a -1 vs +8ish bonus.

The distribution here leads to several unsatisfying results. Realistically, a Monk should be climbing a rope with literally no effort while a wizard should be struggling to even try. The game doesn't reflect this as written. Establishing a DC for climbing a rope is non-trivial because it depends on who is climbing it as to what the intuition should be... again because of the above implementation (i.e. rules) issue. If you have different DCs for different players, you're basically implementing hidden circumstance bonuses from 3e under the covers. If you use your 'fighter' intuition, you get ... DC = 8-ish. If you use your Wizard intuition, you get DC 20-ish. This is a somewhat unavoidable problem given the rules.

It's easier than you think. Even if the chance of failure on the dice is not that different between actors (when it should be) - what ultimately happens on a failure for each actor is completely up to you.
That is a dangerous concept, certainly prone to breaking the suspension of disbelief in a game. Two players roll the same roll and the outcome is different? I can't imagine (barring special abilities) that such an action by a DM wouldn't be construed as favouritism/bias or just bad DMing.