I have read many, many, many articles and watched tons of youtube tutorials on shift registers, but almost every single one only mentions driving LEDs with them.
Same as all the basic/beginner tutorials at arduino.cc. For example, when you learn a new programming language, after it's installed the first thing you do is write a tiny test program that prints 'Hello World' on the screen. Or creates some other easily detectable output. A resistor and an LED is the easiest to construct, cheapest, and fastest way to detect output on an IC pin.
With that you know your basic install of the programming language (and in this case the Arduino GUI) works, and the USB cable and such and you move on from there. In the shift register case the LEDs tells you that everything in the software-hardware chain works.
I have an Arduino motor shield, and other manufacturer motor boards. For test and troubleshooting purposes each output pin has an LED that tells me what that output pin is doing. I operate the board first with no motor attached, and watch the LEDs to give me a better confidence that the motor will work when attached. If the motor doesn't work, but the LEDs 'imply' that it should, then that's a clue.
By my understanding, if you can generate PWM outputs to drive RGB LEDs, you should be able to do the same for motors or servos with, say a ULN2003, to provide the uumph!
2003 has 7 inputs and 7 outputs. That fills up a Uno pretty quickly. One of the easiest ways to create a vast amount of outputs is through only a few pins and a shift register. Cascade them and make more-more outputs.
What I can't understand is why just absolutely no-one ever takes the project further than LEDs, which leaves me with a nagging doubt that I am missing something!
Because that exercise is left to the student. Whatever reason you -need- a mux, these tutorials demonstrate the basic Hello World, and you may proceed from there at will. PWM/stepper/multicolorLED/whatever, whether the data can be processed inside the Arduino bits shifted out fast enough to control 'x' number of motors is a different topic and beyond the scope of any typical Hello World tutorial. All the tutorial does is get you off the ground - you need to decide how hard you need to flap your wings to fly as high as you need/prefer.