Always happy to see great use cases from you :-)
You can add a custom action (that is even potentially visible only for these team members) that will let them choose the task and enter time right away through a form.
The actions in that custom action will be the following:
- add this member as a resource (if he is not a resource already, and for that you can use the function HasRole)
- add a timesheet entity for that user on that task