Router randomly shuts off

Netgear has issues with some of its products lately; for example NETGEAR Cable Modem CM400 has issues with getting intermittent connection last few months ago but I think it was already somehow fixed. If the modem itself is automatically rebootings (as in you see changes on its physical lights) then there could be something wrong with the physical router itself and needs to be changed. And since you already had replaced it with the same one, then it might be best if you get another model.

This answer is very correct. Problem (which I just encountered) and fixed the VERY long way, is that there was a large batch of problematic routers in the field (per the Frontier Tech), being familiar in IT, he was very knowledgeable.
Before he replaced my modem, he did several tests. In all, I ended up with a new "block" on my exterior riser box at road, then a brand new wire from the box to the house, then a brand new router. Still showed errors on the test. So, I ended up with a new exterior box on the house.
After all this, I have well over the 6 meg speed advertised with my DSL, and have had no problems, but it only took him 3 hours to figure through it all and do the work.
The problem I've found is certain Tech's will (only do the minimum), and if it works, even poorly, they're done. Once I called and put in a complaint ticket, they sent a A-1, TOP NOTCH tech, who actually cared and worked through everything to fix "all problems".
As for modem issues, they really don't know which of the ones are bad, it was a huge batch, distributed nationally for Frontier.
Make a call to tech support, explain your issue, and your knowledge of some bad routers (doesn't matter how you know, if asked, just that YOU KNOW).......
Make the complaint in a semi polite, but concerned and becoming unhappy manner, without making it like it is the "phone tech's" fault, and I hope you are given the same experience I got. I'm quite happy now, and FYI, this all took place just yesterday, after 1-1/2 weeks of dealing with it and multiple times of going out to the modem location, unplugging, resetting and then it would work, maybe an hour, maybe 30 minutes, but would still do the on off thing, just as you described.

As for having another device, piggy backed of their (Frontier) modem, I have a highly boosted and extended range D-Link Turbo switch attached to mine, and when I get just far enough from home to lose Netgear wireless connection, say on my android, at the rear of the just over an acre (narrow and deep property), my phone automatically switches to the D-link connection.

Good luck, apology for long post, but I am hoping if you use my tactics, even if they do re-wire your system as they did mine, you will find problem fixed and be as happy as I am.

On a side note, the old wire from riser to house was very dated, now is updated. All other wires were at best, if at all, CAT-5, now all is CAT-6. Not sure how much difference this made, but my wireless speeds on the device side of modem is way better as well.