Pocket doors were popular in houses early in the 20th century. I suspect they went out of style partly for that very reason. You would think smart engineers woulda figured out a way to construct them so that they might be fixable without having to tear out entire walls, or as in this case, major chunks of infrastructure. And evidently you would be wrong.
If you think about it, all you really need is a couple of properly placed small hatches that allow access to those screws. In their absence, you are faced with the problems you have observed.
Pocket doors are a GREAT idea, space saving and all that. But man, are they a major pain in the butt to repair when problems arise. You have my sympathy, OP. And I wish I could give you the secret handshake to easy pocket door maintenance, but I don't know of one.