I truly have not decided what I want to do here, but this article has seriously made me consider dropping late designations all together (having, instead, simply completed work or missing work). If you are interested in weighing in, I ask that you read the article above in its entirety, and then put your opinion here. Moreover, please engage with one another's ideas to help flesh them (and my own thinking) out more fully.
I have a couple ideas in mind that you could also reflect on. First, I would take Dr. Hasinoff's suggestion that there still be due dates, as the course is meant to work at a particular pace--work coming in after a due date would not be penalized but I would require you to visit office hours for feedback. Second, I would also be willing to enter into a binding contract with any student who felt that they needed deadlines--firm, enforceable deadlines--to stay on track. I welcome any other suggestions you might have! (However, to be clear, I have not made any decisions about a change to the current policy as outlined in our syllabus.)
I'm generally in favor of a no-late-work policy, but I personally am someone who really, really needs external deadlines to stay on track. I've got a professor this semester who's got six assignments based on the textbook that he made due on the last day of the semester but that we can complete at any time - that's like, my worst possible setup for getting things done on time. Echoing what the article said about drawbacks - "One student explained that it “gave me too much room to procrastinate,” and another said they had a “hard time finding motivation to complete things.” - I will procrastinate to kingdom come given the opportunity, so to some extent I need to not be given the opportunity.
To me, an ideal solution would be something like a perpetual extension policy - communicate VERY CLEARLY verbally in class and on the syllabus that you'll give a basically endless extension to any assignment /given that/ we requested said extension, say, at least 24 hours in advance (or less?). If you want to make it even easier, the requests could go through a google form or something like that. Obviously not a tested strategy, but I think that would allow students to take time on assignments they need time on for any reason, as well as keep some semblance of deadlines. I don't know, is that the type of happy medium that has the problems of both but the solutions of neither? Not a clue. I don't study educational psychology.