"Free will" is an ambiguous term. The sort of free will you've argued for here could be paraphrased as "not being God's puppets", but I hope it's obvious that that can't be evidence of God's existence. But you listed "free will" as something you can explain with God better than without!

Ah, let me elaborate, then.

Whether God exists or not, one can postulate a universe in which people are puppets - philosophical zombies, moving and acting according to some purely deterministic set of rules.

In the atheistic universe, those behaviours may be at odds with one another, because the rules are not guided; they do not have an aim. They may optimise for some goal on the individual, or even the group level, but there is no reason why they should do so in an efficient manner; a puppet universe may include humans who oppose each other.

In the theistic universe, the presence of an omnipotent, omniscient being suggests that there is some purpose to the universe. If all people are puppets, then, it is to be expected that all people work tirelessly towards a single goal, without opposing each other.

Therefore, the observation that people oppose each other cannot be used to argue for free will in the atheistic universe, but can do so in the theistic universe.

I really don't think the fact that people sometimes do things a god should be expected to disapprove of can be evidence for that god's existence. Do you?

You've got it backwards. I'm not using it as evidence for God's existence; I'm using it as evidence for free will, given the existence of God as a postulate.