For any device, you need an ISP in order to get online. Is your FIOS router tied directly to the incoming service in some way or could you go and get your own? In what way does that FIOS router "suck big time"? Have you considered contacting FIOS support?
I think what you're reading about the Odroid solution is that it BECOMES the router, you still need your Verizon. My brother did a similar thing with a Raspberry Pi, though his solution was for a traffic filter (whole-house ad blocker, porn filter for the kids).