I might have trouble remembering the difference between bubble sort and, say, selection sort off the top of my head. On the other hand, I can implement insertion sort from scratch. It is so simple I do not need to remember the algorithm, I can just figure it out whenever I want.

I think there is a problem with the industry having forgotten the point of why we were asking questions like this in interviews in the first place. The question is not whether you can remember the algorithm, it's whether you can implement such a simple algorithm in any programming language that both the interviewer and interviewee know.

Since the point is the implementation, not the memorization, even simpler algorithms than bubble sort might well make more sense. This is why we've got FizzBuzz for example.

As an alternative, the interviewer could take a moderately simple algorithm like say the Fisher-Yates shuffle algorithm, explain it to the interviewee on the whiteboard, and then see if they can implement it. But again, the point is not to test the interviewee's memorization skills. These algorithms are simple enough that any software engineer who can actually program should be able to implement them from a diagram and some textual/verbal explanation.

The question of coding on a whiteboard vs coding on the computer is a separate question.