I just read into it a little bit. Seems that I was wrong, and “tar paper” is typically not completely waterproof, it’s just water resistant. Most stuff these days isn’t technically tar paper, either, it’s roofing felt. I guess it’s only called paper if it’s made from cellulose. Apparently they did make the felt from asbestos back in the day.

Seems like it’s definitely common enough to use it under flooring, especially if it’s a thinner grade.