The Mystics could simply have commited suicide to rid the world of the Skeksis a 1000 years ago. Certainly that would be quite a sacrifice, but it would have prevented the Skeksis from annihilating the Gelfling culture and from killing countless podlings and what not. And after all the miserable state of the world was the fault of the Urskeks, of which the Mystics had been one half. And the mystics did not seem to care that much when one of them spontaneously combusted.
Or they could have threatened them with suicide, to force them to give up their reign and live in an isolated valley until the next conjunction.
But instead the Mystics decided to leave the world pointlessly suffer under the Skeksis for a 1000 years. That is not that kindly and altruistic.
I think that one point the movie tried to make is that the Mystics lacked any kind of drive or ambition. The Skeksis got those traits when the split occurred. If you think of the old Star Trek episode where Kirk gets split into aggressive- and passive-Kirk, the passive one couldn't make any decisions because he lacked Kirk's aggressiveness. The movie points out that the Mystics lived "in a blur of forgetfulness". They just did their daily routines for its own sake, without passion or reason. The elder even admits that he should have told Jen about the shard a long time ago, but he kept putting it off. Getting rid of the Skeksis, by any means, wouldn't have occurred to them.
Well, they do voyage to the castle and they do their part at the reunification. And the elder does make the effort to raise a Gelfling. And they do keep their secret about being one half of the Urskeks (albeit maybe no one asked). But I see your point. One could wonder why someone like Aughra would not organize a genocide of the Mystics, but maybe no-one on the planet was ruthless enough to do so.
According to books that detail the backgrounds and history not in the movie, Aughra is apparently just a casual observer. She was around before the Skeksis and Mystics appeared and couldn't care less about either of them. She helps Jen, but at the same time, she expressed concern over the old Mystic being close by and, when taken by the Skeksis, implies that she might have helped them if asked nicely, rather than attacking her home. Even if it occurred to her, she is alone. The Gelflings are gone and the Podlings are about as useful in combat as a bag of muffins.
It's also said in the books that the Skeksis weren't that bad in the beginning and they and the Gelflings were friendly, but they degenerated over time. The attack on the Gelflings came as somewhat of a shock, so there was no time to fight back.
For the Mystics, yes, they do get directly involved at the end, but I wonder if that was their will to do so, or if they were simply following the prophesy as numbly as they did their daily rituals. Not so much "We need to fulfill the prophesy and restore balance." as much as "Eh? Oh, look. Prophecy-time. (sigh) Okay, we gotta go, everyone." Just my view, anyway, and it depends on knowledge not given in the movie, so the questions you raise are valid from that perspective.
I think the head Mystic took in and raised Jen out of simple compassion, and realized at the last minute that he'd been chosen to do so for a very specific and important reason (fulfilling the prophecy). They were lazy navel-gazers, yes, but that doesn't preclude kindness. Kind of like, "Oh. This is a pretty pickle we're in. Good thing I remembered to pick up and raise that Gelfling, isn't it?"
What?! WHAT??!?! WHAAAAT!! The Glove is losing its touch!! — Chief Blue Meanie
As long as the prophecy only requires a Gelfling putting the missing piece into the crystal, you would not necessarily need any Mystics/Skeksis to be around. Without the Skeksis you probably would have quite a few Gelflings about, Aughra could just give one of them the shard with the appropriate instructions. There would not be any resistance against the prophesied actions, thus maybe everything would happen alright just by itself. It is a prophecy after all, not a manual.
My impression at the end was that the combined beings left the planet for good, out of guilt for messing with the crystal in the first place, instead of staying as guardians.
But maybe one could argue along the lines of your points that it was necessary that the Skeksis guarded the crystal for 1000 years and kept it for themselves, because due to the corrupting power of the crystal some other evil would have formed instead, requisitioned the crystal and possibly damaged it even more.
Augrah didn't know which one the crystal was. she had it in a bag with some others. when jen asked her for it, she laid them out on the floor, told jen to play his flute/pipes to find it, which he did.