I had "limit access to only assigned tickets" checked on my agent record. That prevented (apparently) from seeing tickets that were transferred into my department since they weren't assigned to me yet, despite me being the department manager. Seems like being the department manager ought to override that, but that's not all that important right now.
Still trying to wrap my head around assign vs transfer though.
Anyhow, that one tweak made everything fall into place. Open tickets are visible by the original manager and the transferred department manager as are closed tickets. Once transferred to my department, I can assign them to an individual, or not. Both sides can see everything. The assignment to an outside team becomes the defacto approval process. Transferring from the team into the team's department serves as the defacto acceptance process. Assignment to an individual is optional (I'm a department of one).
The upcoming version looks very interesting. But I gotta get something deployed yesterday and I think I'm ready to push this out today.
I greatly appreciate everyone's help.
I don't know if this is valuable, but one thing that I might have found useful (and maybe it's out there somewhere) is a discussion of groups, teams and departments. There are several that generally spell things out, but a more granular explanation in terms of where those memberships are checked, which one(s) take precedence, etc. would be very useful in figuring out how things work. Short of looking at the code itself, it's kinda hard to determine. That may be made redundant by the next version though...