After reading your "Blinders" essay, I must say I am struck by how different intuitions can be about a problem. We agree that something is not right about QM, but have gone about looking for the flaw in almost diametrically opposite directions. You have discarded Hilbert space and looked at a nonlinear alternative. I stuck to Hilbert space and looked for a rigorized formal Correspondence. It has been over half a century, but what I vaguely recollect ia that when I included formal analogs of the classical constants of the motion, these attached themselves to the wave function in such a way that by assigning numerical values to them one could sever phase connections. So, instead of everything being phase-connected forever, the equations of motion contained parameters that could cut the connections. That was my approach to improving the theory's relationship to reality.
After all, as long as phases are uninterrupted everything stays in a pure state and is therefore unobservable. Sorry, this is not helping you, since you have chosen a different path. I see it as rather bad news for physics, if nonlinearity is needed from the start. The beauty of QM via formal Correspondence is that it connects directly to the grand Newtonian tradition (through the canonical version of that, due to Hamilton and the rest).