iTunes constantly disconnects from ATV3

I've seen this question posted before but none of the answers have solved my problem.

First, my wifi is excellent. The ATV never disconnects from wifi and plays cloud or internet based stuff without a hitch. I have restarted wifi just in case and it makes no difference.

What happens is that iTunes disconnects entirely from the ATV3 units. Sometimes it does so just whenever. More recently I'll be watching and 5 minutes in it drops it. According to the ATV I have NO sharable computers! Even more recently if I wait three minutes sometimes it reconnects. It is just getting worse and worse to the point that it is unworkable.

I have tried: (1) rebooting the machine with iTunes. Sometimes that brings it back. (2) I have tried turning the machine with iTunes network connection off and on, and/or iTunes sharing off and on. Sometimes that brings it back. (3) One of my wifi routers is not literally 2 inches away from the iTunes machine. No help. (4) I have installed an app called "Keep Drive Spinning" designed to not let my external drive go to sleep (as it has my iTunes on it). That doesn't stop it from disconnecting but when I installed it, that started iTunes reconnecting on its own...so there was some effect.

The machine with iTunes is an iMac running El Capitan 10.11.6. It's a generation 2009 so it's not a youngster but literally everything else works stupendously on it....but not iTunes connectivity.

Any ideas before I attempt to replace it only to find the new machine has the identical problem??

Thanks!