All scientific theories are probabilistic. Just read any scientific paper that reports experiments. There are always probabilities, error bars, p-values, or something to indicate probability. Even classical celestial mechanics, which is commonly thought to be deterministic, is always applied with probabilities. Astronomers estimate the position and other parameters for a planet, and then predict its future position, with probabilities.
Collapse is not really nonlocality. Anytime you make a probabilistic prediction, and then observe a definite value, you are ruling out the other possibilities. In quantum mechanics (QM), this is called collapse of the wave function. It happens in every other scientific theory. It does not mean that nature has any nonlocal properties. It only means that our knowledge of what is possible changes when we obtain new information. The Bayesians say this is essential to all of science.
QM is not inherently probabilistic, any more than any other theory. The wave function is not a probability. What it gives, most directly, is expectation values for observables. If A is an observable (self-adjoint operator), then <ψ|Aψ> is the expectation value of that observable. Usually a range of values is possible, and you can similarly get expectation values for the variance. If you want the probability that a particle is in a certain region, then you calculate the expectation for it being in the region.
QM is probabilistic in the sense that it predicts expectation values and variances, but the same can be said of any other scientific theory. They all predict expectation values and variances.
Any theory can be converted to a many-worlds theory, by rejecting the probabilities. Any time a theory predicts an event with probability p, you could disregard the probability, and say that observing the event splits the universe into many worlds, some with and some without the event.
Many-worlds theory (MWI) is just applying this to QM. It is not useful because the parallel worlds are imaginary, and because we lose the ability to interpret the probability. There is no real notion of being in a high or low probability world.
At this point, the MWI advocates start talking about the measurement problem, or the Bell problem. But MWI does nothing to solve the measurement problem. Measurements do not become less mysterious by postulating that world-splittings go along with them.
Bell's Theorem only says that QM is not a local hidden variable theory. Hossenfelder is wrong to say that Bell proved QM is nonlocal.
The obvious problem with MWI is that it hypothesizes parallel worlds that cannot be observed. There is no scientific value in discussing such things. The main problem is worse. Accepting MWI is the same as denying probability.
Suppose theory predicts a 90% chance of rain tomorrow. That means that if conditions are repeated 10 times, you can expect 9 rain days. But in MWI, it means that it will rain in some of the parallel worlds, and not others. You might think it will rain in 90% of the parallel worlds, but MWI has no way to count the worlds or make a statement like that. You don't know anything about the likelihood of rain in your world, because the MWI view is that there is no such thing. So arguing for many-worlds is the same as saying that the 90% probability is meaningless.
All of the above is mathematical fact, not opinion. Some physicists, like Sean M. Carroll choose to believe in MWI, but only because they choose to believe in unobservable parallel worlds with no scientific value, and because they reject probability theory.
What would Carroll say to defend MWI against these arguments? First, he would say that belief in wave function collapse is unnecessary because it is conjectured that solutions to the Schroedinger equation will exhibit a decoherence that is effectively the same as collapse. I say that may be true, but it does not have anything to do with the fact that the parallel worlds have no scientific value.
Second, he would say that you can still believe in the probabilities, even if they have no direct meaning. I say he has become detached from reality.
MWI takes a perfectly good theory about the world, destroys the part that gives it predictive power, and adds zillions of ghosts that can never be seen. What is the point? There is no practical or computational value to any of it. It is contrary to science. No good has ever come out of it.
MWI is one of those ideas that is so crazy that if anyone advocates it, then his views are on all other subjects are suspect. Other such ideas are determinism, free will denial, and the simulation hypothesis.