"It's not that we can only discover things via science. It's just that we shouldn't trust discoveries made via flashes of insight, clairvoyance, intuition, guesswork, divine inspiration, etc."

If we can't trust discoveries through non-scientific means, than it seems to me you are saying we can only discover things via science. So if I have continuing experience of something so far yet to be confined within an ordered model of falsifiable hypotheses, what am I to do?

I don't understand your question, exactly, so forgive me if my response seems off. But what are you to do? Nothing special. Just refrain from jumping to conclusions. Instead, say, "I don't know." Which is the same thing the ancient Greeks should've done when asked why the sun travels across the sky, rather than assuming that Apollo drags it behind his magical chariot.

Science (I mean physics specifically) is about finding patterns in our conscious experiences. For example, when I drop a rock, I can see it falling.

There are very solid arguments suggesting that consciousness is outside of the scope of current physics. But it's hard to proof that objectively, because consciousness is inherently subjective.