Customizing the home screen?

but I cannot get the home screen to display up next / continue watching.

Up Next is a feature of the TV app, not of the device or the Home Screen. However, an interactive copy of Up Next can be displayed in the “Top Shelf” (above the app tiles) on the Home Screen, when the TV app is the highlighted app in the top row, through an app setting: Settings﹥Apps﹥TV﹥Top Shelf﹥Up Next.

I also bought a gen 2 Siri Remote, in my never ending quest to keep the two TVs as similar to each other as possible, but new remote just will not pair with the TV!

It should pair with the Apple TV box, not with the TV!

Did you try to pair using the remote button combination as described in: If your Siri Remote or Apple TV Remote isn't working - Apple Support

holding down back button & volume up

Yeah, that’s the procedure, hold both down simultaneously for a few seconds. Then a notification should appear on screen.

“If you still need help, contact Apple Support.”

Users may re-arrange the icons on the home screen as they see fit. E.g. move your five most used apps to the first row. You may even group app icons in a folder.

Users may set (program) the TV button on the remote to go the the Home Screen (instead of the TV app), for a fast way back to a familiar starting point.

Select apps allow connecting to the Apple TV app, where shows may additionally be listed in Watch Now, Up Next and included in Apple TV app Search. Other apps did not apply to participate in this Apple TV app integration (e.g. Netflix).

For supporting apps, users may set which apps can connect and which may not.

Users have no influence over what is showcased in the top display inside the Apple TV app, however. That is pushed by Apple. Lots of Apple TV+, and some content from the iTunes Store or connected apps.

Customize the Apple TV Home Screen - Apple Support

Set the TV button destination on Apple TV - Apple Support

Availability of apps that support connecting with the Apple TV app - Apple Support

Adjust Apple TV app settings - Apple Support

Is there any way to 'synch' the two boxes or is there some preference file I can copy from one to the other to keep them identical?

Yes, there is. It’s called One Home Screen, and that’s an iCloud account feature.

Indeed, One Home Screen applies to the Apple TV box, not the Apple TV app.

The Apple TV app can sync its Up Next queue list, though. The Apple TV app can have its Library items synced though.

Other types of syncing doesn’t apply.

Apple TV app vs. Apple TV box - Apple Community.

The last changes on either one will propagate to the other. The configuration (layout etc.) for the home screen is stored in iCloud, linked to the Apple ID account as used for these devices. You may select during setup that the new one should be configured through One Home Screen settings, if I recall correctly.