By the linked argument, introducing any sort of nondeterminism into classical physics would predict many universes.
Not necessarily, objective collapse theories can be non-deterministic without predicting many universes. The extra universes only appear if the wave function never collapses, and stochastic collapses are entirely possible.
If I flip a coin, a classical statistical model would predict I have a 50/50 chance of getting a heads or a tails.
Yes, but critically - under classical mechanics - this is only because you have imperfect knowledge of the system. From the perspective of Laplace's Demon, the result of the flip is 100% deterministic and the chance of it landing the other way is 0. But this is not the case in quantum physics unless a hidden variable theory turns out to be true (and thus any experiment which discovered hidden variables would also falsify MWI)
Do those “other universes” really exist? Or are they simply a figment of my statistical analysis of the situation? That’s the part that’s unfalsifiable.
Well, no. Because you're talking about classical mechanics, where probability is just about imperfect information and isn't part of the underlying ontology. So no, those universes don't really exist. That's completely different from quantum physics, where the wave function actually exists - it's not that the electron only goes through one slit and we just don't know which one: it really does go through both slits.