thank you for the reply. All screensavers are disabled, so it's not that (on TV as well as Apple TV).
I'm currently working with support to diagnose this further.
I also ordered a new remote, because I think the battery burst in the old one...even after I replaced the battery, the old remote won't work.
I also attempted to pair the keyboard off my iMac to the ATV, but it never receives the code seen on my iMac.
Stupidly, while I still had the Remote app working, I removed the ATV from wifi, in the thought that I would reconnect it.
At this point, I have connected the device to iTunes via micro USB, and set it up to Restore.
My new remote should be in tomorrow or Friday, and I am just setting it up from scratch, and will upgrade back to 6.0.1.
After working with Apple Support, the issue appears to be resolved. I briefly put my Apple TV into HLS mode for log gathering, and could not reproduce the issue.
I watched Netflix for over an hour tonight, and the issue did not recur.
Steps I took to resolve this (so that you don't have to relive my pain):
1). Checked all settings on Apple TV to ensure issue wasn't screensavers, power offs, or sleep modes.
2). Checked settings on TV to ensure it wasn't a setting there.
3). Removed ATV from wifi, in the hopes reinstalling it on my network might resolve any communication issues (I had recently upgraded my cable modem to faster speeds). This was likely overkill on my part.
4). Disconnected ATV from power and HDMI, and connected to iTunes on iMac via Micro USB cable and did a Restore.
Step 3 or 4 appears to have fixed the issue, not sure which. In addition, my original remote died, so I had to replace that in the middle of troubleshooting. A bonus fact from Apple: Restoring your 2nd or 3rd Gen ATV doesn't down rev the software version...it stayed at whatever it was prior to Restore. However, anyone on a 1st Gen device is down revved to whatever came from the factory, and you have to reinstall any updates.
All I had to do after setting the device back up was connect to WiFi, re-enter my iCloud information, Turn On Home Sharing, and redo sign ins for apps like Hulu Plus, Netflix, etc.
I hope this helps others going through this issue. Happy viewing everyone.