It sounds to me like you view the technical interview as being split into a high-level whiteboarding phase followed by a separate coding test that is not done on the whiteboard.
In every interview I've had that involved coding and a whiteboard, I was expected to do everything on the whiteboard, from drawing diagrams and working through test cases to writing the actual code under evaluation, all in front of the interviewer.
Under this (very common) interview system, there is no "other piece" for whiteboarding to be more or less stressful than - it's embedded into the entire interview! (Yes, there are other interviews throughout the day that might only involve block diagrams or ask behavioral questions with no coding whatsoever, but I don't think that's what most people have in mind when they're told to prepare for an algorithmic whiteboard interview).
I think the interview process would be a lot better if it started with a no-coding-required whiteboard collaboration phase, followed by coding solo, and then ending with a mini code-review on the whiteboard. This is similar to the private technical interview procedure described in the paper.
As to your other question, I'm not sure why you seem so dead-set against the idea that performing analytical tasks while being evaluated in real-time would be harder than doing so under normal conditions. That's what this paper set out to investigate.