1st Generation AirPort Express - Utility Issue

I have a 1st-generation AirPort Extreme Base Station

You have a 1st Generation AirPort Express if the model number on the side of the device is A1264. If the model number is A1084 or A1088, you have an older "b/g" wireless version of the AirPort Express that was discontinued before the 1st Generation model was introduced in March 2008.

it still isn't showing up as a new device under "Other Wi-Fi Devices."

It won't "show up" in AirPort Utility unless you have the A1264 version of the AirPort Express. If you have the older "b/g" version of the AirPort Express, AirPort Utility on a Mac or iPhone/iPad has not supported this model for 7+ years.

I found this article on The Mac Observer saying that Apple disbanded its wireless team in 2016 and will not longer be creating AirPort base stations.

Apple officially discontinued the AirPort line of routers in April 2018.

is the base station not showing up because Mojave has phased out their support for such an old model? What else could be causing the issue here, and is there anything I can do to make sure this base station is functional before selling it?

Take a look and see what model number of AirPort Express that you have. If you have an A1084 or A1088, the Mac version of AirPort Utility has not supported this version of the Express for 7+ years. Ironically, if you....or a friend.....have a PC, the Windows version of AirPort Utility might be useable with the older versions of the AirPort Express.

More info on the 1st Generation AirPort Express:

More info on the older "b/g" wireless version of the AirPort Express, which was discontinued when the 1st Generation AirPort Express was introduced:

To communicate with it using the AirPort utility, you’ll need to use a Mac with MacOS 10.13 and earlier or iOS 7 and earlier.

This is not correct. Mojave works just fine with a 1st Generation AirPort Express. The OP likely does not really have a 1st Generation AirPort Express. Instead, they have a much older version that was discontinued before the 1st Generation AirPort Express was introduced in March 2008.

The model I'm referencing is a "2nd-generation" MB053LL/A with 802.11a/b/g/n

Any version of AirPort Utility can be used with this model, as long as the firmware on the 2nd Gen AirPort Extreme was updated to at least version 7.5.

If you have reset the AirPort Extreme back to its default settings a few times and it will not appear when you click on Other WiFi Devices in AirPort Utility, the Extreme has likely failed.

Apple has phased out the 1st generation AirPorts. To communicate with it using the AirPort utility, you’ll need to use a Mac with MacOS 10.13 and earlier or iOS 7 and earlier.