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I think my answer to your modified QM is contained in my last sentence above.
The empirically well established wave function that I am referring to is defined in configuration space, and hence nonlocal. If it is used consistently (as in decoherence theory), you may call it “real” (if you like). Indeed, individual states of isolated quantum systems (such as the He-atom – but also Bell states) are completely described by such nonlocal wave functions, while statistical properties occur only in connection with a (true or apparent) collapse. Von Neumann had replaced the kinematical dualism (wave/particle) by a dynamical one.
In the here much discussed Bohmian mechanics, in particular, the wave function is also used consistently (and was hence regarded as real by John Bell). As well known to Bohmians, it therefore also leads to decoherence. Without the thus arising autonomous branches of the wave function, they could not consistently speak of “empty components”. The trajectory is then merely used to define “our quantum world” (a tiny component of the global wave function) as the “occupied” branch. The precise definition of this branch is a matter of convenience. However, since the trajectory is unobservable, this selection by means of a trajectory remains a model-dependent hypothesis; the “subjective selection” of a specific branch from many, many others can be done without using any trajectories. In this – but only in this – sense I agree with the Copenhageners. As the other components would nonetheless “exist” under these assumptions, you are back at Everett!