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By the time you reach a certain point in an academic career, you have accumulated both publications and non-publications. Publications are articles that are published in journals or edited volume or proceedings. Non-publications are articles sitting in a metaphorical desk drawer. They were probably presented at a conference, but not published in a formal proceedings, or sent to a journal and given the dreaded revise and resubmit (R&R), or worse, rejected after a long time. (A quick rejection is a godsend compared to a slow rejection). You and the student both moved on to other projects and, despite your nagging the now former student, the paper never gets revised. Or it gets revised and rejected and resubmitted and then asked for a new revise and resubmit, at which point the now former student effectively abandons it.
From a social point-of-view, this is a useful process IF the papers are truly bad, as it saves other people from wasting their time reading it. However, from a social point-of-view this is an extremely wasteful process IF the papers are not bad, because perfectly fine, if not necessarily brilliant or profound, empirical observations or modeling methods are not published, and thus not cited, not included in review articles or meta-analyses, and thus not in the canon of human knowledge and thus we are collectively impoverished.
There are several problems:
Academics in many fields only cite peer-reviewed work. So simply finishing the paper/thesis/dissertation/report and placing it online in an archive is insufficient to get much visibility most of the time.
Revise and resubmit is a potentially endless process without strong editors. No paper is perfect, there is always more that could be done, and so in the case of Perfect v. Good, Perfect is favored and the paper is rejected. There are enough papers that good journals can be picky. There are enough journals that this paper can easily be resubmitted. Even if it is resubmitted once, it ties up more reviewers and wastes social and human capital. While it might slow down the writing process some, and reconsideration can add value to papers, R&R also slows down the accumulation of knowledge, and more importantly leads to abandoned papers.
Students move on. Most university research is student-driven. A faculty member is a supervisor, may have come up with the basic idea and the funding, and directed the research and edited the paper, but the student did most of the work. If the student becomes an academic themselves, they are often (but not always) properly motivated to revise and resubmit until the paper is accepted. If the student goes into industry, the motivation is weak. Now we could hold the student's degree hostage until publication (I hear this happens in developing countries), but that seems both mean and unreasonable and an attempt to absolve ourselves of the responsibility of determining whether the research is in fact degree-worthy. And it doesn't work if the paper is not in the thesis.
So the solutions:
Don't revise, but resubmit elsewhere and hope for better results. We all know the peer review process is highly random. Nevertheless starting the process over again is not appealing, consumes time, and in any case will require some changes (almost nothing is accepted unmodified, sadly), which the former student has already indicated an unwillingness to do. The likelihood of effort decreases with time and distance from when the work was originally done. Hopefully these changes are not at the analytical level.
Send it to a pay-to-play journal. But if the journal has a bad reputation, the paper will be greatly discounted and not emerge from the black hole of non-citation in which it presently lies.
Edit a Journal Special Issue or Book, and throw the paper in. While this is one way of escaping peer review, it seems a lot of extra work just to get a publication.
Do nothing, let the research whither on the shelf. Maybe someone else will edit a Special Issue or Book and invite you to submit something that turns out to be this very paper, with a light touch for required revisions.
Try to somehow motivate the former students to take the lead. Unfortunately there is always something more urgent (classes, proposal deadlines, conference deadlines, more recent papers that need R&R, health, family, etc.) that sucks away time. There is nothing less urgent than peer reviewed articles in fields which take 2 years from submission to publication.
Change people's perception of the non-peer reviewed literature so that publication is not essential for people to read and cite. I firmly believe the journal system will collapse eventually. But we are not there yet.
I personally am stuck with about dozen of these non-publications which seem to me publishable more or less in current form (excluding papers that need lots of work or are half-finished) but are going nowhere due to the priorities of the lead former student authors, these are included on my working papers page with papers currently under review. They are not coherent enough as a group of papers to stick in a single book (maybe parts of two or three books), and that seems a lot of work, especially given book chapters are discounted relative to journal articles in everyone's perception (and thus their effectiveness). Editing a special issue and loading it with 6 of your own papers (even if each is with a different lead co-author) seems to violate some norm or another, and again the coherence is weak. Editing six special issues and loading each with 2 of your own papers is more socially acceptable, but really a lot of work for the end to be achieved.
So while I don't want to say the world misses out on our brilliance, it is clear human knowledge is poorer as a result of this process.