> However, if Verizon knows your WiFi keys (from the router), it stands to reason they have total control over their routers--so putting your own WiFi behind Verizon's router won't help you if you're looking for privacy. Verizon can see your LAN through their router, even if you disable WiFi.

How does this effect your privacy? If you put your own router between your machines and their device, sure, their device can see the WAN interface & IP address of your router, but it can only access your LAN like every other machine on the Internet at that point, if you have your router do NAT. (Which is the point of doing this.)