Airport Extreme is not visible as a WiFi network with constant amber light. Unable to connect to Airport Utility on the iMac

Unfortunately, a solid, constant amber light is almost always a symptom of an internal power supply failure on the AirPort Extreme.

Try a few more Factory Default Resets to see that might revive the AirPort Extreme.

If you still see the solid amber light, the AirPort Extreme has failed, and will need to be replaced.

As I said, if a reset will not get things working again, the AirPort Extreme has failed.

Apple does not offer repairs on the AirPort Extreme, so when a device fails, there is no other option but to replace it.

Actually this is not strictly true...

My airport extreme base station (bought 2010ish) did exactly this. I bought the current "tower" version and boxed up my Extreme. I'm electronics savvy and I wouldn't put anything in the bin without taking the lid off it. I repaired an ibook on this basis a few years ago. Apple build quality is questionable in my opinion these days.

Anyway. I took my extreme apart. I left it plugged in (its low voltage so no shock risks). everything looked in order (which I expected) but a prod of the battery (yes there is a battery in there) and suddenly the airport appeared as a network. I pulled the power, then re-plugged it and it booted up fine... I set the network up again and it is now giving me internet with no problem.

I cannot possibly say that this was the battery, but I speculate that prodding it helped.

My conclusion is that the battery may play a role in the boot up sequence. If this is the case, then perhaps the battery goes... (its a button battery) and the power fails, then a re-boot may yeild a stall because it may have lost it's settings memory. I have no idea why batteries sometimes get a kick when you prod (or spin them in the case of AA/AAA/C/D batteries)

I must add that this is a guess, but it worked for me and in my case, all was not lost.