Earlier quoted context omitted.
The problem is that it doesn't act like a mass distribution, it acts as two non spatially overlapping possibilities. I think that the only way to save classical gravity would be superdeterminism. If quantum states correspond to anything other than our ignorance, i.e. if superposition states are actual physical states of reality, then gravity will need to be quantum.
I attended a talk by Jonathan Oppenheim a while back on this subject, and my understanding based on that is that in his model something (very) roughly like this happens. You put your massive object in superposition, it interacts with space-time and "tries" to put space-time in superposition but since space-time is fundamentally classical (in his model) what happens is space-time ends up in a probabalistic mixture of…
Well, the question is, can we use gravity in the same way that we use quantum processes, i.e., in a coherent way? If we can, e.g. using it to establish entanglement, then the theory you describe cannot hold.