Airport Utility can't find my Apple Airport Express A1392 after a factory reset

There are three reset 'levels" for the Apple AirPort base stations: soft, hard, or factory default I suggest you try the last one (if you haven't already.)

Ref: How to reset your AirPort base station - Apple Support

if it was me then I would pull the plug on the express and let it start up and if that did not resolve the issue

then I would connect a computer laptop desktop mac or windows does not really matter, by ethernet cable to the express ethernet port and then use airport utility on the computer, then you get a sure connection and you get to see what the amber error code indicates

This also had the result of making it visible to Airport Utility on my iPad.

Maybe a tip for the future.....if you perform a Factory Default Reset on any AirPort router......and.....you want to use an iPad (or iPhone) with the AirPort......you must first do the following on the iOS device......

Tap Settings

Tap WiFi

Tap on Set Up AirPort Base Station

This will open up AirPort Utility to automatically guide you through the setup. Once you have the AirPort set up again, you can tap on AirPort Utility to "see" the settings on the AirPort.

You are correct.