It changes because with ordinary randomness you can’t travel between different branches in the decision tree. In the thought experiment with the surgeon he actually physically travels to a parallel universe and saves a life of his copy there. So the expected long term utility is not 1 life saved but 10 lives saved.
It’s not clear to me that for all observers in our universe, there’d be a distinction between “a surgeon from a parallel universe suddenly appears in our universe, and that surgeon has memories of existing in a universe parallel to the one he now finds himself in.” vs “a surgeon, via random quantum fluctuations, suddenly appears in our universe, and that surgeon has memories of existing in a universe parallel to the one he now finds himself in.”
In your example, rather than consider all infinitely many parallel universes, you chose to consider 10 specific universes where a surgeon appears and “claims” to have come from a parallel universe, and saves copies of himself.
Even in a multiverse where travel between different quantum parallel universes is impossible, you can still find 10 universes where a surgeon appears and “claims” to have come from a parallel universe, and saves copies of himself. You can, in fact, find infinitely many universes where that happens, without requiring any travel between universes.
Perhaps I should have been more specific, I’m talking about a scenario where there is an actual machine (like a time machine but instead of travelling in time you travel between universes) in which you step and press a button, and then you appear in a parallel universe. It’s not a question who claims anything, nor it is a question of random fluctuations, it’s a question of whether that kind of machine can be built or not. If it can be built, then increasing quantum diversification reduces xrisk, because then the travelers can travel around and repopulate other universes.
It is simplest to imagine a scenario where all 10 universes have such machines and you can only travel from one machine to another, so you step into the machine in your universe and you step out of the machine in another universe.
There is also no point in talking about the exact number of such-and-such universes, all that matters is the proportion of the universes in which something happens, there is an infinite number of every possible universe. I talked about 10 of them to simplify the principle, which holds for any n of universes.