Yeah, meat that is “falling off the bone tender” is almost always overcooked to my tastes. I guess some cuts of beef can stand up to that and still be good, but certainly not chicken.
I’ve made chicken broth starting with raw meat, and I’ve always tasted the meat, and then decided to throw it away. I might reserve a little to throw into a chicken soup where it still adds texture and the broth supplies the chicken flavor. But i don’t want to eat it in any dish where it’s a significant component, and certainly not one where it’s expected to taste like chicken.
As a result, i usually make stock from leftover bits of roast chicken.
My absolute favorite, though, is a stock made about half and half from roasted leftovers and fresh raw wings, feet, and backs.