Endgame as in after beating Ganon? After beating Ganon was when I finally started seeing silver varieties of monsters, but I still have yet to see a silver lynel. I've hunted down more of them lately hoping one would spawn after a blood moon, but nothing. Unless it's linked more to the Divine...
Iirc the sidequest is the dragonfly one at one of the stables. I think it was one of the Eldin stables; some girl wants the three different dragonfly varieties.
I find they're easier to come across if you're not actively looking for them. Sometimes you'll just see one fall and you can pursue...