The God of the gaps idea is that since there could be no possible natural explanation, God must have done it. God-as-first-cause is a different argument, because God is the first cause whatever it is, even a natural one.

The fallacy is more one of anthropomorphism: when we think of creation of the universe, we think of a creator deciding to do so (mind), being invested in his creation (loving) and setting up the outcome. It seems clear we have projected our ideas of a parent (our notion of a creator) onto God. Different religions (especially early ones) are the hypotheses that came up in the absence of science, and reflect human biases. In this sense the hypotheses are certainly skewed (I agree the hypotheses are privileged) but not the God-concept itself.

If it's specific enough to include concepts of believers and non-believers and the punishments and rewards due to them

This is why I had added the words ('and fate') up above. It is very, very easy to see design in random events over a lifetime. Over the weekend, a friend told me about how they decided to name their child after a saint whose 'saint day' was a couple weeks before her scheduled C-section. I shared the warm flush of surprise and happiness that her water broke and her son was born on that day after all. (Imagine, God had overseen the naming and birth of that child. What a blessing.) I understand that this fact is the one treasured from hundreds of mundane occurrences -- statistically, this is going to happen sometimes.