I have had Verizon FIOS since they first came out, and have only used Apple Time Capsule, which is also an Airport Extreme base station, as a router with no difficulty at all. The only time that I have used the Verizon router is once when I needed Verizon to ping through to check that signal was getting all the way through to my computer correctly (without a Verizon router, they can only ping to their box outside). It was a simple switch-out. The problem was not router related, but a change in how my computer was configured with them (I started out PPoE, they supported for a while, then went all DHCP).
So, whatever you use, keep the Verizon router accessible.
As a bonus, I have my computer data backed up to, like, 2013.
I use Apple products - YMMV if pc only - but my son is able to wirelessly connect with his pc in the house.