Since my 2013 desktop iMac cannot be updated past Catalina OS, will I simply have to accept random spontaneous shutdowns and reboots?

The shutdowns and reboots are more likely caused by a hardware issue than Catalina.

At this point I suggest taking it to your local Apple Authorized Service Provider for an evaluation, then decide on a repair or replacement Mac.

In order to see if the problem is software related download and run Etrecheck.  Copy and paste the results into your reply. Etrecheck is a diagnostic tool that was developed by one of the most respected users here in the ASC and recommended by Apple Support  to provide a snapshot of the system and help identify the more obvious culprits that can adversely affect a Mac's performance.

Copy the report

and use the Additional Text button to paste the report in your reply.

Then we can evaluate the report and see if we can determine the cause of the problem.