We completely agree. We do all of our interviews via screen share, allowing candidates to use whichever language and environment they're most comfortable with. We also give a selection of problems, one of which is algorithm based, allowing candidates to choose which they prefer.
Ultimately though people are aware they're being watched and assessed under timed conditions so it's going to be somewhat stressful. If we think someone is so nervous they're clearly not able to code at all, we'll offer a take home test as well before making a final decision.
As a matter of interest, does your company also ask managers to go through a management test as part of the hiring process? Let them run a team for twenty minutes or an hour and evaluate them on that basis? If not, why not? :-)
I think "take-home" style tests are the only reasonable way to evaluate this kind of technical competency. They are the closest to realistic. Throw people a small problem of the kind they'll actually be asked to work on. Let 'em deal with it--including documenting their solution!
Ultimately though people are aware they're being watched and assessed under timed conditions so it's going to be somewhat stressful. If we think someone is so nervous they're clearly not able to code at all, we'll offer a take home test as well before making a final decision.