So, I installed the newest tvOS beta, which helped a little. I had been filling with DNS settings, again, helped a little, but switched it back to auto.
Then, yesterday, I discovered my ATV 4 was actually plugged into the router Comcast installed with their security system, not the AirPort Extreme I thought it was plugged into. So it was modem > Comcast router > AirPort Extreme. I plugged it into the Apple router, rebooted the modem, and restarted the ATV. We watched a couple of episodes of the Dolly Parton Heartbreaks (or whatever it's called) in 4K and the picture was back to being beautiful, with no lagging. I was also able to watch the HGTV and Food Network apps with no degradation of picture or lagging.
If you decide to fiddle with DNS settings, make sure to restart the ATV.
Just set a reminder to turn off auto-install of tvOS.