X10 R9000

Wow! I wasn't aware the R9000 was available over 1.5 years ago. I thought it's a lot newer than that. However:

-I have seen numerous postings from the Xbox One S user community that there's an inherent problem with network disconnects with the instant on feature. They've had to keep the Xbox console in energy saver mode to prevent this from occurring, and it appears that's been successful from what I've read.  Have you tried that workaround for your Xbox One S? This is specific only to that device, though. Was instant on enabled before you switched over to Verizon Fios?

-I take it that you've done a factor reset once you've upgraded to the latest firmware revision (you mentioned reboot - little bit of a difference there). If you haven't already done so, it's Advanced -> Backup Settings -> Erase radio button under "Revert to factory default settings."  You'll lose all your custom settings as part of this procedure, though (SSIDs, passwords, etc), so take screenshots and manually reinput, or do a config backup prior to and restore subsequently. If you already have performed this, disregard.

-Are you sure it's only your wired devices that are dropping and not your wireless as well? If you're certain, can you switch one of them (your P/S 4, for instance) to wireless and validate if it still experiences the same network disconnects? I have a P/S 4 connected to the 5G network on mine, and it's stable.

-Is Dynamic QoS enabled? If so, can you disable and monitor for any changes in the overall stability?

-To provide some context, I have an R9000, and it's been very stable (albeit lacking some features), until the latest firmware rev when I enabled QoS. I had to disable same, and it's been behaving itself once more. I have numerous IoS devices connected to it -quite a few wired, including a Smartthings hub, Lutron hub, etc, and no issues with network uptime on those devices. Also, a NetGear switch connected to it that's pretty populated (for work) and no issues there from the R9000 perspective as well.

-The DoS attacks in the router logs appear to be mostly fallacious. To be thorough, though, I would probably vet in your environment to ensure it's not a real problem. For most users including myself, it's pretty misleading info.

-There are also some best practices specific to wireless, but since that's not applicable to your situation, I'll not mention them here.