So what I'm asking for here is how other DM's have dealt with this and how it's affected your games. Is it as broken as it appears? Have your players abused it? If so, what did you do about it?
In my experience, it's not really a big deal, because skill checks are not a big deal in general. Regardless of whether someone succeeds or fails on any given skill check, the narrative is going to move forward and it's unlikely that anyone will die as a result, so it doesn't really
hurt anything if the rogue just
doesn't fail. One thing that I noticed with my rogue is that, even though he had a really high minimum check result, most checks just gave that minimum result, so reliability wasn't even a factor in those few situations where the skill check really
was a big deal. (As an unexpected benefit, most of the checks that the rogue was making involved him acting independently from the group, so not having to roll meant that we could speed through the solo stuff even faster and get right back to the group.)
1. Implement the natural 1 rule for skills. It doesn't seem out of line that a skill attempt, like an attack, could fail on a 1. Even masters of their crafts sometimes err. In fact, I don't really understand why skill checks are exempt from the natural 1/20 rule in the first place. Furthermore, our rogue has the Luck feat, so his odds of failing a skill check would be pretty darn low. But at least there would be SOME chance of failer that would make his roll mean something.
In general, this sort of rule is a bad idea, because it means people should constantly be falling down while walking down the street. One of the most elegant design features of the d20 system is how trivial checks can be ignored
because there's no chance of failure; the DM just needs to determine what the DC is, and then only ask for a roll if they would fail that check on a low result.
If you would always fail on a 1, then you need to roll for every trivial action you take, because you
might roll a 1. It doesn't matter whether your minimum check result would normally beat the DC by 10, because every action has a chance of failure.
2. Change the rule to give him advantage on skills checks with prof bonus skills. Again, this gives him a significant boost to those skills (some of which he also has expertise in) so his success rate would be very high, but again, there would be some small chance of blowing it.
The problem with advantage, just in general, is that it only comes in one degree. Giving advantage automatically would mean there's no benefit to favorable circumstance, having help, magical
foresight, or any of the other things which would normally grant advantage. Rogues also tend to have high bonuses in general, thanks to expertise, so advantage plus expertise would probably let them succeed 98% of the time anyway, and what's the benefit of slowing down gameplay by rolling for something that only has a 2% chance of happening?
And honestly, I'm not worried about the loot he might get. It's the story telling aspect that bothers me. There must be some challenge, some drama, some cost, or what's the point? Isn't the whole point of bounded accuracy that even trivial challenges have some small chance of going sideways?
The point of bounded accuracy is that you don't need to be a specialist in order to succeed. If you look at 3E or Pathfinder, a lot of the high-end checks are in the DC 35 - 40 range, so you need to have a huge bonus to even
attempt them. (Literally, the Escape Artist DC for masterwork manacles is DC 35, and the Disable Device DC for a superior lock is DC 40). Bounded accuracy is supposed to mean that most people can
attempt most checks, which is why
most/I] check DCs are 20 or less; and even a character with a low-ish stat can attempt a difficult check, as long as they have proficiency.
That a specialist might still have a chance to fail on easy checks was never the selling point of bounded accuracy, though it remains in effect in many cases.