Manuscript in pipeline, corresponding author between jobs
Finish it in their own time (that is, if they want to) The only downsides you mention are of a practical nature. I'll explain below why they are not actually relevant.

You don't need an academic affiliation in order to publish in journals. But, I believe you might have a misconception of the affiliations listed on paper: affiliations listed are institutions that provided resources for the job. In particular, while your current employer should in most cases be listed as an affiliation, it doesn't mean that stops when you leave. If part of the work was performed with their resources, they should be listed as an affiliation. In particular, if your performed at least part of the work under their employment, you should list them.

As a special case: if an author (not just the corresponding author, but any author) has a new employer, but the work was entirely done without their resources, it is not an affiliation but they can usually be mentioned as a footnote with the text “Current address”.