I don't think there is an absolute answer here. But generally, you want to start as fresh as possible.

I think the engineer designing the new foundation would have to weigh in on that. Disturbing the soil down to the pier depth might be worse than leaving them, especially since there are possible soils issues.

Ours was a tear down and build new and everything was removed. New grading was done and work in the empty pit before the new foundation was poured etc. We found a spring under the house so that was fun.