In the end D&D is a business, and while the theosophic and pseudo-science feel of the psionic classes have always been cool, a book based on psionics generally didn't sell well. Most D&D rules designers love psionics, which is why it gets updated every cycle, but it generally doesn't get widespread adoption by the average D&D player and is often banned from traditional D&D settings (such as Dragonlance, and the Forgotten Realms) because it doesn't really fit with high fantasy. By recasting the traditional psionic casters as monastic orders (sort of spellcasting monks), they fit slightly better. Psionics with their wierd cystals, and ectoplasm, and far realm wierdness would be hard to incorporate into Dragonlance, and if we are honest with ourselves, we've never really worried about incorporating it. It was only when the monk became a psionic class in 4e did we even give much thought to the possibility that psionics could be a substitute for mysticism at all.
However, you can write all the weird flavour text back in without changing the class at all. You just have to support it on the world-building side of things, rather than having it baked right into the rules. For example, if I wanted to do a campaign with Yak men, alienists, astral projection, magic crystals, etc. etc. I can still use these classes as is, even though the flavour text is more friendly to the narrow high fantasy assumptions of most D&D worlds.
I think the 5e mystic is an excellent replacement for the 5th Age mystic because it covers most of powers that the alteration channeling, meditation, mentalism and sensitivity spheres. I'm not all that keen on the healing powers, because I've always felt mystic healing undercuts Chronicles, but with the recast of mystics as monastic orders, you can simply have the monks and mystics of the pre-cataclysm era be largely religious (Majere/Zivilyn/Gilean) and withdraw with the clerics. With the Age of Mortals, you can have secular or even atheistic mystic orders, with emphasis on inner growth and enlightenment. For necromancy and spiritualism, I am content with the explanation written Dragons of a Lost Star, that it was the dark art of necromancy and it was only accepted as mysticism due to the lies of Takhisis.
Finally, I think this class works well to represent "House Mystic" of Silvanesti. Dragons of Winter Night described some weird telepathy magic wielded by Alhana Starbreeze (why does she have the name Starbreeze anyway?) but never really carried it further. Perhaps house Mystic would be a good place to put the Wu Jen.
The official canon of Dragonlance, for a variety of reasons, is not as good as it could be. I do it better.