What kind of prestige? Peer review is anonymous unpaid work.
A good review does not merely check the correctness of logical arguments, it gives suggestions for the exposition, citing the correct references, putting everything in the right context, etc.
All of the reasons you listed for peer review are valid. The broader point is that peer review can happen outside of academic journals, and nobody has an incentive to submit to them who isn't trying to play the academic prestige game. For a significant example, see the history of Perelman's proof of the Poincaré conjecture.
> What kind of prestige? Peer review is anonymous unpaid work.
Prestige to the reviewed, not to the reviewer.