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This is a very interesting idea. Here are a couple of questions about how it might work:
You write that “it seems natural to assume that the origin of the Born probabilities is the random discontinuous motion of particles”. Certainly some kind of reconciliation of your interpretation of the wave function with the Born rule is required. If a position measurement consisted of sampling the position of the particle at an instant in time, I can see how this might go. But measurements take place over finite intervals of time. How could that kind of process pick out one position (with the relevant probability)?
Second, how is interference handled in your approach. Presumably, in a two-slit context, the particle occupies the top-slit wave packet half the time and the bottom-slit wave packet half the time. But then how is interaction between the wave packets to be explained if the particle is always in either one or the other? The particle presumably can’t interact with itself (as you remark elsewhere). Any suggestions?