airport express won't backup

The AirPort Express will not support a USB hard drive, so not sure exactly "how" you are trying to back up.

Please provide more details on what devices you have and how your network is set up, since the AirPort Express will never appear in the Finder.

I am discovering Apple no longer supports the AirPort, so new devices don't work with it. My new MacBook with Mojave doesn't see my AirPort Extreme either, and won't allow me to use Time Machine to port over my old data using a backup from my old (Late 2010) MacBook Air. I think we're both out of luck.

I am discovering Apple no longer supports the AirPort

This is not correct. The following AirPort models are still supported by Apple:

AirPort Time Capsule 802.11ac

AirPort Extreme 802.11ac

AirPort Express A1392

5th Generation AirPort Extreme

4th Generation Time Capsule

4th Generation AirPort Extreme

so new devices don't work with it.

Again, not correct.

My new MacBook with Mojave doesn't see my AirPort Extreme either, and won't allow me to use Time Machine to port over my old data using a backup from my old (Late 2010) MacBook Air.

Time Machine is not used to transfer data from one Mac to another. Migration Assistant is what you need to use.

How to move your content to a new Mac - Apple Support

If you have a new Mac, you have 90 days of free phone support from Apple. Get with one of the product specialists to have them walk you through the process of transferring data from the Time Machine backups of an old Mac to a new Mac.

Do You have a Apple Time Capsule, or a NAS used as back-up drive via Time-Machine?

Do you have updated your macOS?