Nope, especially in those cases. The more sensitive the data, the more important it is to have records of who accessed or changed it and when.

Huh? I want my bank to be VERY clear on when I transferred money for my home closing (and keep that record). They need to keep records and the browser used, IP used, authentication flow etc. Why does this have to be thrown away?

The personal data itself should not be logged but the event itself should be logged with lots of details that aren't private info.