Transferring games to new device?

Your phone lines and cellular service account have nothing to do with your apps data.

The easiest way to transfer your progress, unless the Apps store progress in a server and you can sign in to them, is to simply backup the iPhone the games are on, and then restore that backup to the iPad when you set it up.

You can backup to either iCloud or iTunes / Finder on a Computer. And then restore the new iPad from there too.

See here: How to back up your iPhone, iPad, and iPod touch - Apple Support

then here: Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

Some App titles can synchronise and store App Data (and Game progress) via iCloud.

Settings > [Your Name / AppleID] > iCloud

Scroll through the list of installed Apps, and ensure that the App(s) that you wish to synchronise are explicitly enabled on both your iPad and iPhone(s). If specific App titles do not appear in the App listing, then they don’t support this feature - but may have their own App-specific cloud service (if this is the case, you should find appropriate account settings from within the App).

If the Apps do not provide the means to backup/synchronise their App data, then you’ll not be able to synchronise or Directly transfer this data. Creating a full device backup - and restoring this entire dataset to another device, will be the only method by which you can transfer this data from one device to another.

I am using my old iPhone 8+ and my new XR to play as two players with unique id. Since my online game refers to Gaming Center to verify my identification and there is no login beyond that, I'm afraid that if I backup my iPhone 8+ and load it onto a new iPad, the new iPad will sync with the XR Game Center and overwrite the player on the 8+.

I started a new player when I got my XR specifically because I was afraid I would lose my status in the game if I tinkered with it.

And I understand my Apple ID has been "simplified" somehow since I bought my 8+ and I wonder if that complicates this at all? My wife has three iPhones -- her current XR plus two older phones, including an iPhone 5 on death's door -- and she is using all three for the same game, different user IDs. Her situation is much more serious than mine as the battery is dying on the 5.

I was pondering buying iPads to transfer some of these games to but I am wondering if that is a fool's errand. Can you advise?

As soon as you've logged into your account,goto my purchases at apple store and all your apps should be there.