the original post said nothing about F
But if you read it in context, its F, not C, not Bb (although I like where you going with that)
even if it was in C, theres no Gmin in C (none whatsoever)
even if it was in Bb, theres no C13 (that would be a secondary dominant or whats commonly called V/V {5 of 5} which means F (if it were Bb)
so you're kinda right on that, but its still F, even if for a few measures, because thats all we got
like I said, bring on the adv theory