My own experience is parallel but puzzling in comparison to yours. I agree-- sick leave does not stop the clock. It is also challenging to get, because the paperwork may or may not be available, and it may or may not be possible to document if diagnoses are in flux so that no specific doctor knows what's wrong or if one is sick enough that it is impossible to do the leg-work for the paperwork one's self.
Cancer is a heck of a problem. Wow. Autoimmune with Covid doesn't sound too great either. You really have been hit with a lot. My sympathies.
In my case, I was also in a position where my advisor was kind of downgrading me. I had been offered an opportunity to go to part time because of some low-grade health problems that were bothering my advisor. I was a bit puzzled by this because I had already published four papers in the research group in my 2.5 years there. However, he was right, there were recent health problems. I accepted the half time accommodation.
What shocked absolutely everybody is that immediately after I went to half time, my health problems spiraled out of control. Essentially the same week. I began having seizures every day all day. Eventually, medication treated this, but in the mean time, I lost my advisor and my funding in my fifth year in the program. This was at University of Minnesota in Fall 2010. I eventually earned an MS in July 2013. One might think that going to part time caused the health problems to spiral out of control but that doesn't necessarily seem to correlate with how the health problems actually work as observed today. It seems to have been a giant unfortunate coincidence, or maybe my advisor and I both had a sense something bad was coming, since I also had been diving deeply into neurology literature thinking something was wrong with me.
I don't particularly have any excellent suggestions. Your backup plan of working as a psychiatrist is a good one. I didn't have a backup career. My partner was unemployed. My family took me in and I COBRAed my health insurance, I believe. Thanks to my family's financial support.
But if you are still coping with cancer and an autoimmune disorder, it may in fact be challenging to work some jobs. I currently am limited in what jobs I can take as well, due to the migraines and seizures that returned in 2016. I was able to finish my 3rd masters degree but I have not been able to take a full time job since. I work part time remotely. I don't recommend "part time remote" as plan A, but if your health is very limiting, it is much better than "totally disabled". I have a lot of sympathy for the situation you are in!!!
In the USA, employers are required to accommodate you, if you can perform the job function with accommodations, but if you can't, they are not required to keep an employee who cannot perform the job function, I believe. I believe. So I'm not quite sure where you stand with regard to that. Research in academia can proceed at a slower pace, but if it is slowed down enough, then other people publish first, or studies that are time sensitive can't be finished and go to waste.
If they are able to freeze your PhD until you're past the cancer at least, that sounds like a great plan. Then you could focus on your health and recover until you're able to return to the time pressure of the PhD.
If it's fundamentally a long term disability, I have no specific suggestions...