You don't need to assign issues to people for them to be able to log time on them. (And you never want multiple assignees on an issue too, but that's another conversation)
Check the project permissions, you should find "log work" is one of the permissions, just make sure your people are included in it.
As an example, I've seen a fix for a case where time was recorded in several places. People needed to fill their timesheets in Jira, and the basic check was "have they done 40 hours this week", but their actual time on some tasks was recorded elsewhere. So to pass the checks, they had a single issue called something like "logged in system X" where they just logged a single line every week for the total they'd logged in more detail in X. It was assigned to the timesheet checking person as someone needed to own it, and the project permission said "Log work: any logged in user".
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