Nope, I've been really stupid here. I have a blind spot on this particular problem, and I forget it EVERY time it comes up.
I am really sorry.
The advice to turn off "allow unassigned issues" is correct, but you can't make Assignee mandatory with field configurations. Your only option is to put some validators into the workflow. Which is not pretty for the users.
Basically, it says that you need to use a third-party plug-in: JIRA Suite Utilities
Field configurations, or turn off "allow unassigned issues" in global configuration.
There is documentation on this stuff you know, and it's searchable - see https://confluence.atlassian.com/display/JIRA/JIRA+Documentation
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