But if you give them something in addition to Stunning Strike, which you just said is really good, then you just end up with something too good.
However, if you replace Stunning Strike, you just make them worse, because this new ability would, presumably, work primarily, if not exclusively on spellcasters.
Spellcasters are plenty dangerous, but A: They aren't going to be in every adventuring day, let alone every encounter, and B: That sort of thing wouldn't fit with the fiction of the base Monk. Monks are about being the class that sees a Dragon and says "Catch these hands.", not stopping spellcasters. That and having a base class whose primary identity is basically being a satellite, as in, orbiting around something else and pretty much doing only that, is dumb.
It's a given that Stunning Strike needs to go.
I don't think there's much chance of it surviving into 1D&D in its current form. Because it works on absolutely everyone and stops absolutely everything
So I'm looking at what I'd consider the only realistic option. Replacing Stunning Strike. Also I'd say ranged units in general should be given issues by a Monk, ideally.
As for Monks telling a dragon to "catch these hands", well we could redesign Monks to be that class, and that would also work, but right now they're Mr Chain Stun and nothing else, and because that's powerful in certain situations, they can't
become anything else. I was under the impression you wanted to keep the skirmisher design, though. I guess not?
Really? I didn't know a skirmisher was something that was primarily supposed to deal with spellcasters. I thought it was something that ran up to the main force, pestering and weakening them, so that their main force could come in and have a much easier time of doing things.
I guess that's one meaning of skirmisher, the sort of "screening force" but it doesn't make a whole lot of sense in this context. A monk who "runs up to the main force" of enemies is like to immediately get downed. Their AC isn't that good and they don't have that many HP. Patient Defense only helps so much! You can't certainly can't screen like the cavalry or javelin skirmishers you're seeming describing.
A more conventional approach to a skirmisher
in this sense would be meaning a unit which AVOIDS the main force of enemies, and attacks where they're weak, particularly attacking units not really capable of defending themselves, like archers/slingers, artillery, and so on. That's where very high mobility is extremely handy even where the unit isn't hugely damaging.
The problem Monks have is they don't do enough damage to like single-handedly wipe out a bunch of these guys, and that Stunning Strike is actually better applied locking down some big melee-hitter than on this backline guys. So what's even the point of their mobility? You seem to be suggesting basically they can run in and
kill steal help the other PCs by abandoning whatever they were doing in favour of applying their DPR to a low-health target, but buddy, that's a job that ANY RANGED CHARACTER can do better in 5E. Sheesh, a STR Fighter with some javelins likely does that as well. There's no need for personal mobility to do that when ranged works as well as it does in 5E.