Pros and Cons of assigning homework for large classes
Given that your job is teaching, not just lecturing and grading, I don't see any cons. If the students don't practice, they won't learn. People learn through practice and repetition, not from hearing lectures and taking exams, cramming from notes and a book. 

Learning something involves physically rewiring the brain. The neurons get new connections (synapses) that need to be reinforced. See The Art of Changing the Brain by James E Zull for the science behind this. 

Sadly enough, students don't really know how to learn in too many cases, nor do they have the time or discipline to do what is not required of them. So you need to give them meaningful practice as well as a goad to get them to do it. I think, actually, you would be more effective if the percentage of the grade for homework was much higher than 10%. 

If you have adequate support (TAs) then there should be little reason not to have such graded homework with comments not just numbers on them. 

If students get a lot of practice they will actually remember things and be able to put it to work later. Otherwise it is as fleeting as the end of the course. 

Even if it is impossible to grade all the work you give them, give them the assignments anyway. While it doesn't solve the goad problem, it at least gives guidance those with a bit of discipline. Even if you permit them to work together on homework or to grade each other's work, it is better than expecting them to learn without practice, practice, practice.