That's kind of completely not how D&D works, though. Like, you're welcome to run it that way, and I'm sympathetic to running it that way, but that is absolutely not RAW or RAI. Particularly not the different DCs. DCs, in D&D, are not set "to the character". That would make for a completely different, wildly different system.
So basically you're literally ignoring the rules, and using DM fiat to plaster a pretty obviously wonky issue that is the direct result of using a d20 for these rolls, combined with having a relatively small bonus to the roll (even for the "master"). Again, I'm sympathetic to that, but claiming that somehow "proves Saelorn wrong" is absolute and total arrant nonsense.
Yes, if you ignore the rules, and use DM fiat, you can fix almost any issue with any system. But that's what you're going to have to do.
The reality is that isn't much of an issue in an actual play, though, simply because in 5E (and indeed most versions of D&D except 3.XE/PF), this scenario just doesn't come up very much. 5E particularly makes it very unlikely due to the "Working Together" rules, which means the master almost certainly has someone else in the party helping him, which means he's making the check with Advantage.
But please don't tell people the system isn't wonky here. This is a specific issue with all systems that use a d20 and a small bonus as the main roll.
Whilst they're obviously papering over a hole and claiming there is no hole, which is ludicrous, it's not true to say anything with 20 points warrants a check. I mean, you can run it that way, but that's also not really RAW or RAI - though less severely, because you're not making up rules, you're just overusing them.
The moderating factor is generally if something is fairly easy, you shouldn't require a check, even if it theoretically has a DC of 5 or 10 or whatever.
In reality, because of the way 5E works, and the Working Together rules, this is rarely an issue - but it's certainly an underlying flaw, and ignoring it or pretending it's not a flaw is pretty bloody silly. It's something that becomes obvious as a flaw the moment the players forget the Working Together rules and start making separate checks on something like Arcana - because of the high variance and low bonus, especially at lower levels, you'll frequently have a character who it seems it is um... unlikely... would know that who actually makes the check. The difference between 10 INT, no skill, and 16 INT, skill is only +5, so the odds of the other guy knowing rather than the wizard are actually pretty good. If D&D used 3d6 or something it'd be a completely different story, of course. But it doesn't. Which is cool. But we can't pretend the issue isn't there - it's just rarely a big deal.