Have to perform a hard reset with each update, to turn console on

It's been an issue for some people since a recent update. Try plugging the controller into the console and performing a hard reset (hold the power button down for 10 seconds until the console turns off, unplug the power cord for the console, plug it back
in and turn it back on)

Have you tried performing a hard reset on the console yet?

If that doesn't work, you can always try the offline update for the console

Both have Gold and have each other's profile set as the other home xbox? Do you get any error messages? If so what are they. Have you tried hard resetting each console?

Try performing a hard reset EXACTLY like this. Hold the power button on the console for 8 seconds or until the power shuts down completely.
Unplug the power cord from the back of the console for 5 minutes.
Plug the power cord back in and restart the console.

Anytime I have ever run into this error, I would perform a hard reset of the console. Hold the power button on the console itself for 30 seconds. After that, turn the console back on and that should have worked.

I have fixed this problem with a hard reset. For anyone that doesn't know how to perform a hard reset...

hold the xbox light on the physical console till the machine turns off. Then wait 30 seconds ( recommend by support article). Power the console back on, it will take a minute or two to complete the boot process.