That's quite a list, thanks for sharing! I must admit though, that with 20+ AppleTVs a lot of the problems you described just seem to be exactly what I would expect in that environment, and I hardly doubt that you will find a more perfect solution than the one you've already figured out.
What I mean is that AppleTVs are home devices for the living room, and as a necessary means of action from Apples point of view they are designed to work exactly in that environment: the living room. Or the bed room. Maybe another one in the childrens room.
But for the professional and high demanding environement you (and I further above) are describing, I would simply suggest looking for a professional solution that's made for this environment. We can and will figure out a way to work around Apple's home design, but it will only get us so far.
A software product that we are using and which has brought us better results is the AirServer application:
https://www.airserver.com/ With good server/pc design (positioning, network design and so on) you might find it better suited.
EDIT: I am not working for airserver
😄 after reading the post I just had to point that out.