We would like the ability to assign an incident to be separate from the permission to manage it. i.e., a T1 can reassign one of their tickets to a T2, but they can't see or edit the T2's tickets.
This is possible with permissions. You need to restrict T1 from seeing tickets assigned to T2. They will still be able to re-assign tickets that are currently assigned to T1 and change it to be assigned to T2.
That is true only in the case that you don't want the T1 to see the ticket. A separated permission would allow you to do this regardless of whether they have Read access or not. Another issue you may encounter with that solution is a T1 CC'd on a ticket now having access to edit it even when not the assignee.
