How realistic is this wishlist?
Probably not if all the requirements are dialed up all the way to 10.
Very realistic if you're OK with some of them ending up at 6 and you still make a perfectly usable, but more importantly UNIQUE paper.
Do it. I'll follow your progress with great interest. I've pondered the same route, but haven't gotten round to it yet. Never went beyond some desk study, which went into the direction of fermenting nettles and separating the fiber from them. For gluing/sizing several options are available; I'd probably start with plain old gelatin. It has worked for centuries; it'll still work today.
Doesn't have alkaline buffering which interferes with a lot of alternative processes
Should be no problem at all; just don't add a buffer. Maybe your paper will deteriorate faster as a result. It probably will. Who cares!?
Has a neutral tone and a reasonably bright white
That's going to be a tricky with natural/self-harvested fibers one since most fibers will be somewhat tan to downright dark/brown. Perhaps give it a try with something 'silly' like plain cotton from the store? That's pretty white as it comes.
Retains some amount of tooth/texture
I think achieving the opposite would be far more challenging; i.e. making a hot-pressed paper will be very challenging. But a coarse paper should be feasible.
Can withstand being submerged in various trays of wet chemistry without disintegrating
My guesstimate is that if you (1) use decently long fibers and (2) your sizing is sufficiently hardened, you should be OK for stuff like cyanotype. Processes like carbon transfer that require hot processing will be more challenging to make decent papers for. The long fibers should be easy to cover with stuff like cotton. Sizing shouldn't also be too difficult; gelatin hardened with something like glyoxal, formalin or even alum should do just fine. You can also see if you can get some
AKD which is one of the go-to agents in the papermaking industry.
Doesn't require expensive specialized equipment to make
I'd look into manuals that explain how to make screens for screen printing. I have a feeling that should get you plenty close enough to making a papermaking screen.
Just some thoughts from an armchair theorist. I hope someone with actual experience chimes in.
Again: do it!