John 10:33 is typically translated as something like

"“We are not stoning You for any good work,” said the Jews, “but for blasphemy, because You, who are a man, declare Yourself to be God.”" (Berean Study Bible)

Jesus responds to this by talking about the gods of Psalm 82.

"Jesus answered them, "Is it not written in your Law, 'I have said you are "gods"'?"

This seems like a non sequitur. According to the standard translation, they are accusing him of being God and Jesus is responding about gods - God's representatives. How is that an adequate response?

This suggests another possibility, that the Jews instead said something like

"“We are not stoning You for any good work,” said the Jews, “but for blasphemy, because You, who are a man, declare Yourself to be a god.”"

This would make some logical sense of Jesus' response. This would also fit with what Jesus says at John 10:36.

"How then can you accuse Me of blasphemy for stating that I am the Son of God?"

Jesus is saying they are accusing him of blasphemy for being the Son of God - not God Himself.

Is it grammar that is primarily driving the typical translation of John 10:33 which renders the Jews' statement as 'God' instead of 'god', or are background theological views driving the translation?