Update to the above: it looks like the issue is that the tomcat server is not auto-starting, i.e. not starting as a service or process at startup.
If I manually kickstart it by going to /Library/Filemaker Server/Admin/admin-master-tomcat/bin/ and then running ./startup.sh, tomcat/catalina/coyote will run and the Admin Console has something to connect to. Then everything works.
However that requires a manual start of tomcat with each reboot -- and even then, the process runs as an application that must remain active in the foreground.
Command-line management (fmsadmin) works fine.
Note that I've uninstalled and reinstalled multiple times already.
Even better solution:
I called Filemaker this morning and they had posted this "patch" one hour before I called. (It's not really a patch, you have to edit a config file but it's easy enough.) The whole thing is indeed just a problem with catalina not autostarting -- the fix is to add a Java switch instructing catalina to operate in "headless" mode...
I wish I'd known this before I deinstalled and reinstalled FMS. I had to recreate all my schedules etc.