Yea, the difference between a classical statistical theory with and without many worlds is whether or not you maintain a state that includes all possible outcomes as you continue your analysis
Yes, but, again, this is only because you have imperfect information about the underlying physical system. The array of possibilities presented by classical statistics are strictly epistemic; the actual real state of the system you're analyzing is always definitive and determinate.
And very, very importantly, this is not that case in quantum physics. The indeterminacy of state in a super position is not just the result of imperfect information; it is a fundamental part of the underlying system. It is not the case that, in the double slit experiment, the electron only travels through one slit, and we just don't know which one. It really really does travel through both. This is fundamentally different from classical mechanics.
I’m arguing that quantum mechanics is a rethinking of statistics more so than a rethinking of physics.
Look, if you want to try and argue that quantum physics isn't physics, I won't stop you, but you'd better have an extraordinary argument, because this is an extraordinary claim. One that rejects the last century of scientific consensus. If you can demonstrate that quantum mechanics is just a different statistical model of classical physics, it would be a revolution in science.
I think that even if I must consider a state that includes all possible outcomes while doing my analysis of the situation, that doesn’t mean those “alternate worlds” necessarily physically exist in any meaningful way.
That's correct, and MWI doesn't argue otherwise. The important part isn't just that these states are possible, it's that they have real physical existence.