No one really knows how Janeway would have dealt with the Prophets. I suspect that this thread is nothing more than another attempt on how terrible she is as a Starfleet captain. And probably because she is a woman. Once again, Trek fandom's bigotry rear its ugly head. Why on earth didn't you ask this damn question about Picard, Kirk or Archer?
I chose Janeway, not because she is a woman, but because she is the only captain who in practice rarely, if ever has to defer to / trust in "higher authorities" in the series, so we don't get to see this side of her character much. (Picard, Sisko, Kirk and Archer all have their superiors who they have to answer to, Janeway in practice doesn't (at least not during the run of the series)). On rare occasions, we see a tiny sliver of it - for example there's that episode
Sacred Ground where she realises in the end that the guides were right: the ritual
is meaningless, it was only she herself who needed to "invest" it with meaning, and that it would have been a lot easier on her if she simply could have accepted what they told her instead of going the hard route with her preconceptions. I don't like the episode itself that much, but this is a side of her I wish had been explored somewhat more during the series' run.
Sisko learns to trust (to some extent) in beings that are beyond him in scope and comprehension. I can't for the life of me see Picard ever doing this - he will defer to authority but only if he understands it himself-, so that's why I did not chose him for consideration - this doesn't seem like an interesting area to explore. Janeway however, just might. Then again, she might not. I have to admit that I did not think further than these three contemporaries.
However, I agree that Kirk might be an interesting idea: how would he deal with God-like entities that for once aren't a scam, nor wish to be left alone, nor really try to exploit him --but still manipulate him when it suits their agenda, which usually seems to be for a good cause even if not always understood ? Would he have been able to establish a "working relationship" with them? I genuinely don't know. Archer I somehow can't visualise interacting with them at all.
Is this a response to the Punch It podcast?
Never heard of it, no idea what it's about (but I'll look it up one of these days).