Seems to me you just shouldn't ask typical "CS problems". The problem with those is that a positive result doesn't necessarily mean that the candidate knows anything. It could be that they've just interviewed so many times that they know a few common answers to memorize.

I still think asking coding questions is extremely important. Programming is one job where you can actually test skill in the interview (unlike management for example, where you mostly have to rely on personality, work history, and references). I don't know why you woulnd't take advantage of that opportunity.

As for culture fit, obviously you need to do that too, but that's somewhat orthogonal to what else you ask about in the interview.