"What does psionics do for the themes in Dark Sun outside of just being psionics?" What it does is be psionics. It's not mundane and not magic. It's job is to be fantastic and not magic.
For you.
For me, Dark Sun never needed psionics and we almost never used 'it'. Heck, I don't even remember if any of our characters had psionics or wild talents. It never was the focus of any of our stories in DS.
But that's just me. It's totally legitimate that for any other group, psionics were what made DS for them.
Dark Sun requires Psionics. Without it, all you have is a post apocalyptic setting that is similar to Dark Sun. That can be fun, but it's not going to be Dark Sun.
Sure, but at that point you have some other post-apocalyptic setting. Not Dark Sun. You can create any number of cool post-apocalyptic settings, Dark Sun being just one of them, and one which heavily involves psionics.
See, this intolerance for allowing anything to be accepted as Dark Sun except exactly what a specific person/group wants is exactly why, imo, WotC should never republish Dark Sun.
I believe the designers Wizard's have are truly capable of creating something new, different and if not better at least as good as anything we've had in the past. And re-publishing anything, no matter how tolerant I am, will split some of the community. Such will never bring us together, but only continues to split the community. And, to me, is lazy to fall back on what was done before. Yes, I feel the same way about Eberron, Saltmarsh, DoMM, etc.
I would much rather leave DS, Greyhawk, SpellJammer, etc to the past. They all have more than enough content for people to play for generations and the fans can and have updated them to their tastes just as well as Wizard's can do.
What Wizards can do that almost no one else can do is to create new settings, new storylines, new adventures that do not call upon the past, but invent what is new. And they can do this because they are the gorilla, when they publish something, it reaches a market like no other, it is adopted like no third party publisher can do. What they publish, defines the genre. Let's not define it the same it was 20 years ago.