The PHB describes the Weave as an interface with the “raw magic” that exists in all things. I will be very disappointed if psionic characters need the Weave at all. On the other hand, I would have no problem with psionics as an alternative way to harness this raw magic. That is exactly what psionics as presented in 5e is (and it has been presented in 5e since the core books); another way of harnessing/manipulating the weave.

That’s where the arcane vs divine distinction lives.
Magic is magic, what differs is how you manipulate it.

— Dan Dillon 👥 (@Dan_Dillon_1) December 22, 2019