Why is it 'of course impossible'? Couldn't we someday have nanobots or other tech that could screen all your cells and be able to indicate whether any cancerous cells remain?

If we were able to find cancer cells with a precision down to 1 cell, killing them would be trivial. There are lots of cells in a person. It's possible we will get there someday. I think bioengineering is the most likely route. I.e., design a virus to specifically target the type of cancer cell you mean to eradicate.

Strictly speaking, no impossible from our understanding of physics, but also an enormous scientific and engineering problem. research into that would take away from people working on more reasonable approaches today.

No. It's hard to get even small molecules where we want them in the body, there's no way that the gigantic molecular clusters called "nanobots" could reliably get access to every single cell. (What may be possible is using things like the Moderna cancer vaccine to make your body an inhospitable environment for the growth and multiplication of the cancer cells.)