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
It's not perfect, but allow for create issue with unassigned users. Assuming your tickets go to the backlog, it's not a big deal as most backlog items are not being worked on and are generally unassigned.
Then, to transition to via Start Work, or In Progress, place a validator on the Assignee field. The user will get a red error message the Assignee is required.
This makes assignee required to start work on a JIRA configured for unassigned issues.
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