MacBook Air early 2014 from Big Surf 11 to a version that will accept my apple ID

That's a new one. I never heard of an Apple ID that "can only be used on devices running iOS 16.2 or later, or macOS 13.1 or later."

Google search turned up this thread:

Why can't I log into my Apple ID with Mac… - Apple Community

Based on that thread, I'm guessing that you turned on Advanced Data Protection for your Apple ID. Especially since the iOS and macOS requirements in that error message are the ones in this Support article:

How to turn on Advanced Data Protection for iCloud - Apple Support

Since your Mac can't run macOS 13.1 (Ventura) or later, it logically follows that you "Can't Use Your Apple ID on This Device." (At least not unless you disable Advanced Data Protection.)

For the Early 2014 MacBook Airs, Big Sur is the end of the line. You can't upgrade them to Monterey or later.

What do you mean about Big Sur not accepting your Apple ID? If Big Sur won't accept your Apple ID, that's likely to be due to some problem other than the fact that you're running that particular version of macOS.