It may be squeezing air bubbles or many things. The only point I've suggested is that it does not involve proper classes. Most of the axioms of ZF are assertions that there exist certain kinds of sets. The Grothendieck axiom is another such assertion. That it is not entailed by the axioms of ZF doesn't refute that it is a way to get the category of sets without resorting to proper classes. That's all I've claimed.

I don't ask anyone to adopt any axioms they don't care to adopt. My only point is that there is an axiomatization that provides for the category of sets without involving proper classes.

And that's the only sense I suggested.