A strong single-case design should ideally involve multiple baselines (across subjects, setting, or behaviors), multiple treatment conditions, measurement of distal as well proximal outcomes, logging of external events, etc, as well as statistical analysis of results. If people are passing off, say, progress monitoring as an acceptable dissertation study, then just no, period. That being said, I work with a lot of SCR people (my [R1] department has probably one of the most prominent collections of SCR methodology faculty in the country), and all of the grad students I know who have focused on SCR have done meta-analyses of SCR articles for their dissertations. I don't know if the faculty would actually accept a single-case research study as a full diss or not, tbh.
FWIW, I've been involved with and published both group and single case research, and originally was pretty dismissive of SCR for the reasons many people are. I have to say, I have a lot more respect for SCR after seeing what truly methodologically rigorous SCR looks like, and also seeing that respected and well-trained SCR scholars look at individual SCR studies not as conclusive statements but more as steps forward to meta-analysis after a sufficient number of studies have been conducted and published. I'll probably always be a group researcher (or meta-analyzer 😉 ) first, but I think well-done single case research can and does make significant contributions to knowledge, especially when examined meta-analytically across studies.
Most people I've known have done correlational studies for their dissertations--I've only known two who actually did RCTs for their dissertations, and only one was a tx RCT, with a fairly small n (about 60). Tbh, while I love treatment RCTs, I really don't think they are practical for a diss due to the time and money required.The only way I could see one realistically working would either be a small pilot RCT, latching onto an advisor's already in-progress RCT, or having your only grant to do one and a willing to take a year or two extra (if not more!) to graduate. I always encourage people to use pre-collected data if they can, as I've known people who got incredibly delayed by recruitment issues. To paraphrase one of my mentors, your dissertation should be a solid, strong study, not the capstone of your career.