Note that most people are taught about the idea of a “universal set” that contains everything.
This Is A Bad Idea.
The farther away from a “universal set” you get, the happier you’ll be.
Jabba’s proof of why we cannot have a sets x and y each containing the other is valid, but not satisfying. What you have shown is that the existence of x and y is not allowed by the Axiom of Regularity, not how the existence of x and y (in the absence of AoR) would create a Russell-like paradox.
I’m kinda ambivalent on this issue. For the non-mathematician, it really doesn’t matter that much. But yeah, once you get into any sort of advanced math, you need to do away with it.