Bathtub lift bucket is missing.
I only get to work with one type drain stopper, its all metal, so I don't know about the rope you pulled up. I can only assume if the parts broke off, they would be blocking the trap, in which case I'd try a bent coat hanger down the overflow pipe. I'd also try a plunger to get "stuff" moving around some. Although I don't like using drain cleaners, sometimes you need to.
A ridgid brand "sea snake camera" would be helpful although its hard to know what you are seeing. Keep us informed how you get through this problem.
LIVING THE DREAM