That appears to be a conceptual objection, rather than a scientific one. Not gonna refute a theory that way.. But it raises a problem that nobody, AFAIK, has ever been able to solve: "the probability of me being one of the me's that sees a live cat" doesn't make sense, because both of the branches after the measurement are "you". "You" before the measurement does not become one or the other branch after the measurement, with some chance of each; "you" before the measurement becomes both branches after the measurement. This is the issue I referred to before as the problem of explaining where the Born rule comes from in the MWI.
Anyway, we don't need to go down this path. The equations work. The wave function evolves deterministically regardless of which interpretation one takes. Determinism and probabilities is not an oxymoron. The probabilities most likely (no pun intended) represent our knowledge of the system. As long as it obeys the laws of physics, I can sleep well at night.