Well, I don't have the old router set up in the configuration I described above...at least not yet. It's worth pointing out that I've *never* had rebooting issues with it, however. Not once over the course over four years.
This was a final test to see if there's anything physical on my end to blame. Seems pretty clear after replacing the ONT, the router, coax and splitters that there isn't. I'm convinced that the problem is either the router, Verizon's network or both.
Interesting to see what tier 3 has to say about this tomorrow.