How do the consequences of Russell's paradox extend beyond universal comprehension principle as far as the set of all sets problem?

Russell's paradox does not in itself prevent a set of all sets from existing. There are set theories that do contain a universal set and are not known to be inconsistent, such as Quine's NF.

It is only together with Zermelo's subset selection axiom (the core idea behind ZFC, which claims that $\{x\in A\mid \varphi(x)\}$ is a set whenever $A$ is), that it has this effect. If a universal set existed, then the subset selection axiom would effectively provide a universal comprehension principle, and then Russell's paradox would produce a contradiction.

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it is often said that Russell showed, with this paradox, that the "set of all sets" does not exist.

While it is probably not difficult to find a claim such as this in print, it is an ahistoric oversimplification. Russell published the paradox several years before Zermelo proposed his axiom system, so the ingredients for making the jump from "unrestricted set comprehension doesn't work" to the specific claim "there is no universal set" was not present at that time.