How is that relevant to the choice of whether or not to have homework impact the final grade in a course though? I left research and whatnot out of the equation because, to my eyes, they really do seem unrelated to the observed pattern of professors intentionally choosing to use homework as part of the final course grade.

If anything, if I'm focusing on research, I'd prefer to not waste time grading a bunch of assignments. If the students aren't going to read my responses, why write them? If they're all cheating and going to get 100% on the homework anyway, just add the +20% to the homework blank at the end of the course and be done with it.