Power through it? You only have to do it once (or four times if you respond to all four), it takes one day and it's not that hard. There are way worse and fruitless ways to spend a day that we do routinely.

For a lot of skilled engineers with cushy jobs, not going for whiteboard is matter of principle.

I understand, but the question was on how to overcome the unwillingness. I don't see any other way to overcome it than 'just do it'. It is not a physical addiction or a condition and bears no ill effects other than lost time and maybe some pride?