Why is a "good alchemy system" different from a general magic system? Why don't we just refluff sword attacks as fire bolts and have nothing but fighters?
Because mechanics inform the fiction, and if psionics is the same as wizardry you haven't come close tto the trope of "psychic powers."
Huh? In 5e, it's the artificer that has an "alchemy" system and it's pretty much straight up standard casting with a couple of tweaks. So, are Alchemists not flavorful enough? Are they just wizards?
As I've said, regardless of how much psionics is like magic, if you reskin a caster as a psionic, you are intentionally making your character less powerful/versatile to fit a theme. It just doesn't feel good. Psionics should at least have it's own class/classes with it's own abilities, even if it uses the magic system.
I have no real problem with psionic characters getting a new class. That's fine. A new class with three or four subclasses? Sure, I can buy that. Psionics are certainly flavorful enough to make a good go of that. It's the "new magic system that doesn't interact with the existing one" that I draw the line at.
For example, if we go with the "psionics is totally different" angle, that means that a psionic character can't use any of the "caster" items. No wands (by and large) or various other magic items. Which means now I have to have a whole suite of magic items just for this one class, many of which are just replications of existing items - so, no Pearl of Power, but, a Crystal of Mind Power, things like that - and that's a non-starter for me. And, of course, if we have this entirely new system for this class, do we then have to apply it to monsters too? Again, this is a non-starter for me. Too much work for a single class.
My basic argument boils down to a pragmatic one. If adding psionics to the game requires more than, say, about six pages (which is what the artificer takes) or so, then I'm just not interested. It's too complicated, for me, and too much work.