That means it picks a branch at random, which contradictions Many Worlds' premise that the universe is fully deterministic.

Or it picks the most convenient one. I don't know what this word 'random' means. If what you mean is 'arbitrary', there are lots of reasons you might arbitrarily pick this or that.

To claim you continue living in another branch is equivalent to claiming your experience somehow jumps

For one, I don't actually care. But for two, most depictions I've seen portray this as a passive effect. It is not you that moves, it is the universe that moves around you. The details of how this actually works, or doesn't, is for nerds in comic book stores to discuss.

I'm just cautioning people to not think quantum physics actually does support the idea,

At the risk of sounding a bit argumentative—and I really don't mean to—Who?