How do I setup a backup 4G/LTE network?

The Ethernet switch, managed or not, is not the issue, you will still want one for what you are trying to do. A non-managed switch should work just fine for "manual" reconfiguration of your network. However, there is no "automatic" switchover here. Also, I've never heard of an Ethernet "Y" cable to do what you are suggesting with going without a switch.

Each of your modems would "initialize" with the device that it is directly connected to it. Basically, the modems will remember the MAC hardware address of that device. In theory, you should be able to power-off one modem, and then, power-up the other to use it with your AirPort Extreme.

However, to make sure that they communicate properly, you should also power-off the Extreme BEFORE switching modems. This is somewhat analogous to powering-down your networking hardware at night and powering them back up in the morning.

Is there no way of putting something (a switch of some kind?) between the modems and the Airport Extreme (that would take both WAN feeds but put out only one)?

In theory, perhaps. But this would require a programmer to write a custom program for a managed switch that would ignore the signal feed from the 4G router unless the feed from the main Internet connection was not available.

is it possible to use the Airport Expresses to extend a non-Apple Wireless router’s network.

Not possible if you are asking about having the AirPort Expresses wirelessly extend a third party network. Possible if the AirPort Expresses will be permanently connected back to the main router using a permanent wired Ethernet cable connection.

I wonder if I could get around this being too complicated (in programming terms) by only powering-up the 4G router if the other is unable to reach the Internet.

As long as you are going to manually power up the 4 G router, it would only take another 5 seconds to disconnect one Ethernet cable from the WAN port on the AirPort and replace it with another. Of course, you will want to power off the AirPort Extreme before you move the Ethernet cable, then restart the AirPort.

You could use a simple Ethernet switch if you don't want to swap Ethernet cables. Do not use a splitter. Power off the AirPort Extreme again as above. Power up the 4 G router, then power up the Extreme.

It's stating the obvious that none of this meets your original requirement of having things happen automatically.

I found some old GUI scripting suggestions, but I assume there's no command-line option that I am overlooking?

This is more in Tesserax's wheelhouse, but I would note that it is possible to restart the AirPort Extreme from the AirPort Utility application, so software does exist for this. That being the case, it might be possible to write a separate script for this.

I have to say the complexity of what you would need to do simply to keep Apple Airports in action as main router.. shows they are unsuitable once you exceed their very domestic usage profile.

Almost any $200+ router today includes failover to USB 3g/4g modem.. eg Asus.

You can still run the Apple routers as WAP.. although the improvement in signal that the Asus gives might eliminate the need for a lot of those Express.

To do what you ask, you would need to replace the AirPort Extreme with another router than has two WAN ports.....something from Cisco, for example.