Assume that many-worlds interpretation is correct and quantum immortality is true. Essentially, in at least one universe you survive no matter what dangerous things you try.

These two statements are not equivalent. Quantum immortality refers to the hypothetical (supposedly inevitable) subjective experience of ever-continuing life, the latter sentence seems to describe the continuing existence (in atleast one branch) of some person you could currently consider to be a future version of your current self.

I believe in the latter, but not in the former. There's nothing mystical and immortal about my self-perception of continuity, or memory, that it should persist in the branch that "I" follow; even if we limit ourselves to the branches where there's some continuing qualia-producing processes, why should the sense of self be the process that remains functioning last?