Earlier quoted context omitted.
You can have optional assignments and quizzes that serve that purpose. If you take away the credit given for homework, you still can give feedback, while removing any incentive to cheat.
Few students do optional assignments unfortunately. Other tasks that are directly worth a gradetend to take priority (e.g. studying for another class that has an exam this week).
2. I had a prof. that didn't require homework be done. He would give out "practice fun" and would gladly sit down, give feedback and 1:1 time to those who completed it, or tried. He also pointed out that it was rare to pass the exams for students who didn't do "practice fun". Most people did the work.
It leads me to believe - from my own experience too - that students generally aren't stupid, and will gladly do the work if there is a point. Plenty of homework is pure busywork though, even at the college level.