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.