The interviewers are ideally trying to get a sense of how you think through problems, not just that you can spit out an answer you know. At least that's what they say.

So does struggle equate to working through a problem? I don't think it does.

It's not the struggle, it's the questions you ask, and what you share about your thought process trying to got to a solution. You might work yourself into a hole solving what you think is a simple array manipulation problem, but then realize some edge cases require you to switch to a heap/priority queue, etc.

Not really, no.

What if you happen to be an unlucky genius like J. von Neumann or something?

You still should need to ask clarifying questions, because the specification of many problems is intentionally incomplete.

Beyond that though, you can just be honest and say you know a problem and they can either pick another or just talk about it anyway.