I would disagree with your professor. Cheaters don't just harm themselves. They harm other students. Consider that there are a finite number of seats for university admission, and a finite number of scholarships and jobs which are decided upon by merit by way of grades.
Cheaters essentially jump ahead in rank above others that worked hard for their grades. This pushes the honest students further down the rank. Arguably, the worse hit students are those honest students at the bottom of the rank who are completely displaced from their opportunities. Those students (even though we and they don't know which) should be reasonably protected from losing opportunities they rightfully earned.
There are many other issues with cheating including changing the moral norm. Consider that cheating by faking a foul in basketball is just part of the game. Does that mean it's ok? Or consider that some couples cheat on each other thinking that it's ok as long as they don't get caught. I've met many people who think this way nowadays. Personally, I'd like children to grow up in a society where this does not become the norm.
An increasing minority of students feel that cheating in academia is ok if you don't get caught. They view academia as a game with referees (instructors) that determine fairness. If the referee doesn't do anything, the game will not be fair. I fully understand that we can't catch all cheaters but we should try to do what we can to minimize the problem.
I would say that it makes the job of an instructor extra burdensome and in many cases untenable when the administration is not on board as well. Maybe your professor is in this situation but doesn't explicitly state it. It's been rather difficult to deal with at my school. I do what I can and I know other instructors that do as well, however, I don't see much support from administration. I see the problem getting much worse.
Access to information is continuing to get easier as resources and technology evolve. I don't see how universities can solve the cheating problem or slow its increasing use without instituting some standard of academic integrity through proctoring of some kind so that we can verify identities and work. I'm not sure how this can be done or if it's even possible.
Just my opinion. I hope it worked out.