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The router is NOT a modem. The handoff from the ONT on FiOS is a direct ethernet connection (either over Cat5 or Coax via MoCA). If you have FiOS TV, then you must use the router in order to all the STB's to talk to the internet for guide data and to allow any of the remote programming services (such as DVR remote) to function.
If you don't have FiOS TV, you can ask Vz to provision your ONT for ethernet and then connect a router of your choosing to the ethernet connection on the ONT. You will be responsible for running the connection from the ONT to wherever your router is located.
If you have FiOS TV and don't wish to use the features on the Verizon provided router, my suggestion is that you configure a router of your own choosing with it's WAN interface connected to a LAN port on the Verizon router, statically configure it for an IP on the 192.168.1.x network (and use a different network range on the local LAN side of your router), and then login to the Vz router and disable wireless and place the router IP you selected into the DMZ on the Vz router. This doesn't eliminate the Vz router, but it effectively routes all traffic into and out of the router to your router and lets you do everything from there.
As for the Vz router itself ... I have no problems with mine and have had it for several years. Perhaps you could be more specific about what you mean when you say "touchy".
What is the DMZ of the Vz router? I want to use my own router as well. The verizon one is just a pile. Locks up if too many computers are on it. I get most of your instructions.
Just so I'm clear.
1. Connect Lan port on vz to Wan on new router.
2. change vz router to static IP of 192.168.1.x
3. change new router to 192.168.2.x
4. login to vz router and disable wireless and place new router IP into DMZ on vz router?