It shouldn't be necessary.
To confirm upfront:
Then, once you've provided the user(s) with Assign Issues permission:
I tested this on Cloud without edit permissions and it worked fine. You can't inline edit the assignee as this does class as "editing" the issue - but the assign button is the workaround to that.
Based on Omar's account which you've logged into - it appears he can both
He won't be able to inline edit as this is editing even for Assignee.
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