One thing I don't think has been mentioned is setting a DC can be subjective if you want to play that way. It solves most issues anyone might have with Bounded Accuracy IMO.
Scenario 1:
Take the blacksmith NPC. For him, crafting a sword might be a simple task. Sure, it takes
time but he knows the process and has done it many times during his life so the DC could be 5 or even 10, and then add in a STR 14 (+2 mod, strong, but not a Conan-type) and "expertise" in Smith's Tools for another +4 to +6. Even at a +6 total with a DC 10, he might mess up 15% of the time--meaning most attempts will succeed but sometimes the quenching is off or whatever and the sword is weaker than it should be. It might still even function, but perhaps will break on a natural 1?
Now, look at a PC who has STR 18 (+4 mod, more the Conan-type) but no proficiency. Sure, in 5E part of that +4 might represent some "training" in
everything STR-related (a very laughable concept and to me the REAL problem with 5E), including smithing a sword. The DM could rule without proficiency, the task is understandably very difficult DC 25 and there is no chance the PC could do it. Even at a more reasonable DC 20, there is only a 25% chance of success without proficiency.
Scenario 2:
A street performer NPC is playing his flute for the crowd, playing a simple but pleasing melody he wrote. He is reasonably charismatic with CHR 14 and has "expertise" in Performance (which, btw, means he can effectively sing any time of music, dance any style, play any instrument he picks up, etc. -- again, laughable and over simplistic IMO but ok I'll just go with that) and like the smith NPC might have a total +6. For him such activity is routine and might warrant a DC 5 or 10.
But, the PC Warlock with CHR 18 decides to try to make a little money for the party by singing a simple melody to go along with the street performer. Not having proficiency in Performance, her "training" from CHR 18 still grants a +4 bonus. But because she doesn't normally sing in front of a crowd (a scary thing for most people--even heroes) or practice singing much (if she did, she
would be proficient in Performance), the DM sets the DC at 20. Why? Because failure has consequences (or at least setbacks!) as the crowd boos and hisses at her failed performance. If the player rolls low, the DM might start with some taunting and allow the player to roll again, maybe even with disadvantage. A couple lucky rolls and the PC wins over the crowd, but it is most likely downhill and rotten vegetables.
The summary is this: depending on the situation and just who is attempting the task, the DC could be easier or harder. Some people might not like playing that way (which is fine), insisting that a simple melody should be the same DC no matter what, that crafting a sword should always be the same DC again no matter who is attempting it.
Another idea I had to reflect on this idea was to simply impose disadvantage on any check where proficiency in a skill is applicable but the PC doesn't have it. While it helps, it doesn't create enough of a difference between the people involved in scenarios like those above. In the smith example, the NPC would be 85% with DC 10, but the with the same DC 10, the PC would be over 56% chance of success (better than 75% RAW, but not enough).
Maybe reintroducing a "non-proficiency" penalty would be a good way to resolve it, like -5 or even -10? That way, you can assign the same DC for the people in the scenarios, but the smith's DC 10 will succeed 85% (using +6) while the NPC's would only be 25% with a -10 penalty and 50% with a -5.
Just more to think about.
