How to investigate why students do not do their homework?
You have been around the block a few times, so I interpret this question to be "What is going on with THESE PARTICULAR students?" And of course I don't know.
I have run into some scenarios at my large public university which may be relevant:
Recent admission changes which bring in students from more low-income schools with less experience with self-regulated learning.
Changes in course requirements which allow less-experienced students to take the course.
Limited timing of a different required remedial class made one of my two sections also all remedial.
A shift to making a class more online also allowed students to fall behind more easily.
Whatever the reasons, I would also want to know why this results in homework avoidance and how to help overcome this.
Have you tried having undergraduate tutors ask the students what would help? Your students might feel safer admitting problems to peers.
There are many aspects to this question, which amount to "what are some good teaching techniques?" In fact, I think getting students to do homework assignments is one of the most important parts of teaching. I recommend you look for in person training programs at your intitution's (or a nearby institution's) teaching center. Each institution in this network has such a program: http://www.cirtl.net/.
When I did this, I learned how to design better assignments, lead better class discussions, create more engaging classes, and find out what determines student success. I think most senior faculty would benefit from similar training.
In this case, I suggest creating in-class activities the students cannot do unless they did the homework.
Also, in my experience, all LMS suck. Don't use them except when you have to (typically to keep grades confidential).