Nope. That's where the fundamental duality of concepts and other things chips in. A set is a concept, not a thing. It is in people's mind. Likewise numbers. And it's trivially true that we can invent an infinity of concepts. There's no limit to concepts. Infinity is in this sense our creation: a mere concept that we use in many different and useful ways.
But there also exist (at least in a realist dualist woldview) things outside our minds, which people call 'real', for lack of a better word ('objectively real'? 'non-conceptual'?). Things like real apples, real molecules, real particules-cum-waves... Those 'real' things are modeled, represented, signified by our concepts. That's one way to use concepts at least. But I cannot eat the concept of an apple the way I can eat a 'real' apple, so there is an obvious difference between the concept and the thing it represents.
So far this is trivial. Back to our discussion: we can prove logically that a "set of all sets" cannot be conceived of, because sets are mathematical concepts we can manipulate logically. It does NOT follow from there, that 'God' -- the 'real' creator of this 'real' universe -- does not exist.
My point is rather that your conceptualisation
of God as "the set of all sets" may be a nice metaphor, a beautiful poetic expression. What it CANNOT be is a logically coherent conceptualisation of 'God'.