I don't think homework should be abolished, but its weight in grades should be limited.
I couldn't agree more. When a large chunk of a student's grade (say 20% or more) comes from homework, its purpose can be defeated. I think of homework as being something that helps a student learn a new idea. If the homework is graded (e.g. 8/10 = 80%), it's like the kids have to be perfect or nearly so from the moment an idea is introduced in order to get an A or B in the class. IMO, classtime and homework are for learning, and tests are the time for grades that count (when I taught, I always gave the final exam the most weight).
I took a first-year history class once that required a short essay every Friday. Each essay received comments and a "grade" but the grade didn't count. It was to help the students track their progress. All that counted was that you made an honest effort and turned it in. The point was that you were supposed to learn how to write and learn how to use sources.
