The question you're asking has certain logical affinities to Spinoza's ontology. Basically, Spinoza's argument goes something like this:
- "Everything that exists has a certain essential property or depends on a certain kind of
power which I'll call NN"
- "When I say say God I'm referring to NN"
This is congruent to the argument you suggest, if you substitute "mathematical describability" for NN. What Spinoza uses for his NN is something closer to persistence and has two attributes, extension and thought (mathematical describability would arise out of the correspondence between thought and extension).
This idea wasn't wholly novel to Spinoza; the Arab philosopher Averroes had a similar idea, and he was building off of different ideas that had been tested out by Aristotle, the Stoics, and Neoplatonism. The primary worry about this type of argument, historically, was that it was "fatalist": it implied God was totally passive/indifferent/uninvolved in particular events in the universe; He might make general rules like "don't murder your friends" that typically would carry bad consequences but if someone murdered his best friend and suffered no consequences, there was no sense in which a fatalist could claim God cared about that. The technical language: there may be "general providence" but no "special providence". (It also creates difficulties for the question of whether God can inspire people, or reveal truths to them, and implies some version of determinism.)
But the coherence of this distinction depended in odd ways on classical metaphysical categories and it wasn't clear if this problem still mattered in Spinoza's system (or if the special/general distinction still made sense). Spinoza himself wrote as though he took the Bible seriously (and lived on a sort of Mennonite compound in rural Holland), but many critics thought Spinoza was a heretic, his piety was a sham, and identifying God with the universe amounted to the same thing as saying there was no God at all. Others thought that, heretic or not, Spinoza was entirely sincere, in fact a "man drunk on God", whose divine will and power he saw everywhere.
Yet eventually the objection almost everyone could eventually agree on was that nothing Spinoza had said excluded the possibility that the God he had described was a blind, mindless, meaningless force. (This summary is unfair to Spinoza: on his own terms, he had specifically shown that God was a substance that had unlimited mind as an attribute, and thus was hardly "mindless" - you'll have to study if you want details.) So the question becomes: granted that perhaps some entity, or more technically some substance exists that we could call God, can we also say that this entity is a subject (a person or agent that is active or purposive in some sense comparable to human beings, who have "life stories", characters, make choices, and so on)?
Hopefully that resolves your question. Some people do try to answer the question in this way; it has difficulties; as a result, some people avoid the argument you suggest because they think it doesn't prove the existence of God (or the right kind of God) at all, and the people who find it attractive (including me) typically, after spending a lot of time familiarizing themselves with classical attempts to understand its texture, jump right over it to the bigger challenge of understanding how God is a subject.
(I should also mention, since you are an atheist and undoubtedly interested in the atheist rejoinder, that the most common atheist objection to arguments akin to Spinoza's is that they do not have much in common with what ordinary people mean when they use the word "God", and therefore are not talking about God at all. The problem with that response is that dumb people generally don't have preconceptions that will withstand careful philosophical scrutiny about anything - you wouldn't normally refuse to consider a sophisticated atomic theory, or construction of the real numbers or something, because ordinary people have a less sophisticated, inaccurate view of electrons, arithmetic, etc.)