This is a tough situation when the district has blanket policies that are negatively affecting students. We want more differentiation in instruction, but these types of blanket policies are disruptive to that line of thinking. I love this post from Cult of Pedagogy- a discussion about different aspects of late work policies- https://www.cultofpedagogy.com/late-work/.
Personally, I allow students to turn in make up work and late work any time up to my final deadline about a week before the semester ends. I too, was originally panicked that this would cause more work for me and that students would develop bad habits. Guess what!? The students that always turned their work in on time, continue to do so. The students that never turned their work in on time continue to do so! The difference is I can work with them on an individual basis to determine why they are having the issues that they are having, but the responsiblity goes back on their shoulders to make sure they get their work in and talk to me. It's no longer my problem to assess each student's lateness and keep track in the gradebook- which is exhausting as a teacher and takes time away from more important matters- like teaching! The students that are able to overcome those issues, turn in their work and get a grade that reflects their mastery of the content, not a grade that reflects the issues that originally caused the late work. It has been a win-win.