There was a George Fawkes baptised 9 Jun 1822 Hickling with parents John & Ann.
I have an instance where a chap remarried and first marriage named a father but the second one didn't so there can be other explanations as you say.
I would try and trace this George just in case and yours did have a son John.
Yes, there are one or two potentials, but pinning him down is the problem. I have no idea, for example, what surname he might have been baptised under - my experience is that it is generally the mother's name unless the clergy are being particularly pointed about the reputed father, and I don't know the mothers name for sure.
I say I don't know 'for sure' because Ancestry throws up a DNA match for me with descendants of a woman who was married to someone else at the time and continued to be married to them for some time afterwards. That could also have other interpretations, but it adds uncertainty into the mix. I don't know, for example, if he
was actually baptised.
Although unlikely, I'm exploring every avenue I can in case, for example, clergy have added a marginal note anywhere.