The Genie 2 does has a wireless video bridge built into it. There is no way to control that feature in the DTV menus. But it operates on the 5GHz band only.

No idea if this could be an issue for nearby WiFi (I actively try to get people off of DirecTV these days; so I don't deal much with their BS anymore). But since it doesn't broadcast an SSID, you will need a scanner like Metageek to view any interference in the spectrum.

To the point of your problem: It does NOT sound like a WiFi issue. The DTV clients (which are not wireless) losing connection, could be the result of IP conflicts. Which would explain all your symptoms.
Again DTV have become some unruly devices and absolutely hate dealing with them on a network. Is the Eero performing the DHCP server in this setup? Any other devices that could act as a router/DHCP server in the system? Do you have a DHCP range set? What does an IP scan of the network show?

Lastly, does Eero allow you to do subnets? If you are not controlling the DirecTV system thru IP, I would isolate it in its own subnet.