I disagree. They report a correlation without a cause, then conclude it confirms the safety. (continue reading the Discussion section you claim I didn’t read).
The conclusion is “there is no increased risk for mortality”. A more accurate conclusion would be any increased mortality due to the vaccine is less than the natural difference in mortality between the two cohorts. Unless they are claiming the mRNA somehow only helps with mortality for non-covid deaths (or has no effect — zero additional deaths due to the jab).
I think you're right and I think their conclusion is flawed. With such a massive difference between the two groups to start with, they can't claim to know with any certainty that the vaccine doesn't at least slightly increase mortality. Maybe the difference between the two groups was 82% before vaccination and 72% after. Even though that's unlikely it can;t be ruled out on the evidence so far.
But that does not mean that they claimed that "once vaccinated, your all-cause mortality drops 72%". That is simply a lie and spreading it weakens the credibility of critics who point out real problems with the so-called vaccines. It's exactly this kind of nonsense that the fact checkers love to disprove, while they ignore the good arguments.