I currently have FiOS. I've never used their router for WiFi. For years I used their router plus various WiFi bridges.
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.
For this and many other reasons, a while back I pulled Cat6 from the ONT (optical network terminal--where the fiber comes into your home) to my office. I'm using an ASUS RT-AC66U for a router. I'm running DD-WRT on it instead of ASUS' firmware.
Unfortunately, all of their TV STBs need Internet (for the guide), and those can only get Internet via MoCA (i.e., Internet over cable TV coax). (Also, various services Verizon offers will not work with the BYOD router setup.) You can purchase MoCA bridges, but it was easier to just neuter the Verizon router and use it as a MoCA bridge (behind my router). By doing so I am probably opening up my LAN to spyware again. Then again, if the STBs are on your LAN, they could be spying on you as well.
I keep an eye on the STB traffic, but who really knows.
I think my next step will be to curtail, then discontinue FiOS' TV service. Lately I seem to pay more for less (their STB interface is frozen in time, slow, and now has advertising to add insult to injury). I think the equivalent or less money will buy sufficient entertainment from Internet streaming or digital downloads. And as a bonus there's less potential for Verizon LAN abuse from their STBs.