How do I use AirPort Time Capsule as an extended Wi-Fi base station on an existing network when the current iteration of AirPort Utility will not detect existing networks to join/extend?

"Extend a wireless network" is a proprietary setting among Apple routers, designed to only let an Apple router wirelessly extend the signal from another Apple router.

So, if you want the Time Caspule to "extend a wireless network", the network will need to be provided by another Apple Airport router.

Another way to say the samet thing........the "extend a wireless network" setting is not compatible with a 3rd party wireless router.

The Express can "join" virtually any wireless network for service with AirPlay, for which it was specifically designed. But, the Express does not "extend" or "repeat" the WiFi signal when it joins a third party network.

But, the AirPort Extreme and AirPort Time Capsule do not provide AirPlay functions. That is why they are not designed to "join" a network at all.

Like the other AirPorts, the Express can "extend" or "repeat" the WiFi signal from another AirPort router. It was not "extending" your previous 3rd party network(s), even though you might have thought that it was.

There are little to no standards in WiFi having to do with repeating a signal, so manufacturers are left to their own devices to come with their own ways to extend the signal.