Apple TV (headless) + Home Sharing + Remote stops playback after one song
I suspect your problem is the network, not the devices. The queue used when controlling the AppleTV from an iOS device is different from the queue on the native Music app (to see this yourself, note the empty queue in the AppleTV music app when you start music from your iPhone).

So, you have two dependent links (at some frequency, not necessarily constant):


iPhone to AppleTV, to update the queue, provide new information, etc.
AppleTV to iMac, to retrieve requested files


Can you monitor the network links on these devices using your WAPs diagnostics features? Or, perhaps more usefully, using Wireshark or the like to promiscuously capture the conversations? I'm willing to bet there's some severance going on.

Regarding running the AppleTV headless, I'm certain that's not the anticipated use of the device, but I'm not aware of any specific problems this could cause, unless it expects video signal via HDMI to keep alive, which you could test easily enough.

Hopefully, this is somewhat helpful in considering your problem. Good luck!