I'm all for code walk-throughs and explanations. You can't bullshit your problem solving process, but you can memorize (or forget) algorithm trivia.
For what it's worth I've failed and succeeded at these kinds of algorithms on the whiteboard interviews. Even after nailing it I felt uncomfortable and disrespected. I think there is a real problem with continuing to do these interviews because it's the cool thing to do, and these smart people don't feel like turning their problem solving skills towards the hiring problem.
What if someone asks a friend for help or pays someone to solve the problem in its entirety? If your problems aren't unique enough, they could also google the answers.
With that said, I actually prefer your approach. Its pitfalls have to be weighed carefully; hiring someone who knew enough to fake it could be a costly setback for a small company. However, it does remove some unnecessary pressure from the interview process.
Note that algorithms interviews are just one part of a larger inventory that one takes of a candidates, one dimension. Take home and/or whiteboard coding can be a part of that process as well, but each has its downsides.
What we look for, in particular, is methods that have a very low false positive rate. In person algorithm interviews, where the point is not the 'right' answer but hearing about how someone thinks are in that category. You can't really fake it.