I've got two of these horn button "cuffs", both are pretty firm.. almost plastic feeling. They've not been exposed to elements, but maybe aged out. One has a little more give than the other. Are these supposed to be soft, pliable rubber? Or are they usable like this?

You can try the most pliable one and see how it goes. Not difficult to swap it back out for a new one if it doesn't work.
https://914rubber.com/steering-wheel-horn-rubber-cuff

Are those cuffs only needed for stock steering wheels ? I have a MOMO wheel and don’t recall seeing one…?

They are on a 911, 912, 914 plus style factory wheel same as I have in my 6. The rubber part works like a spring to return the horn from the tone.

Bob B