But its not contrived -- its like your entire job. You're literally saying "I can't perform my primary job function when there is any pressure whatsoever". That MIGHT be acceptable if you're interviewing for an entry level job.
The only other reason I can think for being unwilling to do this is if you're incapable of working with other people - because whiteboarding out solutions with other devs is super common. Even "where do you think I should start, I'm a bit nervous and an idea to get me going would be great" is a perfectly acceptable question. The best interviews are a conversation -- if you turn it into a conversation where we're working together on the problem, it shows a lot about you still. If you take someone else's idea and run with it I think that's really cool, obviously if you need hand-holding it's suboptimal, but there is still a very positive possible outcome from saying "I have no idea where to start" if you're actually nervous.
The only ways I can think of to fail writing out code is people skills or being completely unable to program. It's not like I expect a perfect solution in the 30-60 minutes we have, analysis paralysis is just as bad as no-knock raiding the problem. There is some type of middle ground that involves a good solution that had to be done quickly -- maybe you even mention some other avenues to explore.
What if you ever have to represent my department in any capacity and someone asks you what your ideas for a problem and you just say "no, I don't do that" -- that doesn't really represent me well. I would be very surprised if being uppity about a question being "below you" ever works, you might as well just walk out the door if you're going to refuse to answer a question on "philosophical principle".
The question is just a jumping off point -- its up to you to turn it into a conversation that shows me how great of a programmer you are. If your answer is telling me how dumb my interview is, then that doesn't show me how great a programmer you are at all -- it shows me you have an attitude problem.