To my eye, it looks like multiple questions are being asked at once, so I'm going to try to pick them out and answer them:
Is quantum mechanics deterministic, or probabilistically random? You'll read a lot of opinions online, but the reality is, this is still UP FOR DEBATE among experts. There are many different interpretations of quantum mechanics, and they all pretty much make the same predictions, but they have different ideas of what's going on "under the hood" so to speak. I would say the three big schools of thought are probably Copenhagen, within which randomness is probably real, Many Worlds, which is deterministic but which appears effectively random to a person within the system - at the level of human experience, it's random, but it has a sort of 'meta-determinism' - and the third is Bohmian Mechanics, which is just straight-forwardly deterministic (or at least most people consider it deterministic, some argue it's not). Copenhagen is probably the most popular, but that's arguably because it's taught by default since it was kinda the first intepretation. There are more interpretations than these 3, but these are the most popular it seems among experts.
Can quantum randomness "average out" to macroscopic events being effectively determined? YES, certainly over small timescales, but not over large timescales. You asked if WWII would start at the same time if we restarted the universe - if quantum randomness is real, then the answer is "if we restarted the universe, WWII probably wouldn't even happen, human beings would likely not exist at all". But if instead of rewinding to the beginning of the universe, we instead rewound to the beginning of 1939 (that's when it started, right?) then I think there's an argument to be made that the probablity is pretty high things would play out pretty close to the same way at the macro level, so that it does start on the same day.
We know quantum randomness can average out to macroscopic "determinism"-like situations, because that's why our macroscopic experience is so consistent to begin with. It's why as you're looking at this screen, you can reliably see the words I've typed instead of seeing random noise - because the quantum randomness that's present in every single photon arriving at your eyes averages out to a consistent image. If your eyes relied in any individual photon arriving in the correct way from the correct place, vision wouldn't work, but vision is a bit more robust than that - because it instead relies on millions and millions of photons accumulating - that's how they have the opportunity to "average out" across all their randomness.
Here's a little write-up on the topic: https://www.askamathematician.com/2012/05/q-is-quantum-randomness-ever-large-enough-to-be-noticed/