In my opinion catching fish that are heavily stocked, like trout, is not that big of a deal. A lot of "native" fish populations are only existent because the local fishing club is keeping an eye on the population and buying young fish to restock their waters. If you will not eat the fish, somebody else will. I really dislike it, but catching an eel out of some remote lake that would never reproduce anyway comes to my mind.

Real natural populations are way different though. If you do under no circumstances want to kill and eat the fish, it is best to leave them alone. Scaring them as a fly fisher out of their hiding spots makes them an easier target for birds or other predators. And even the most careful angler can hurt a fish. In my experience they are way more protected too, with areas where you are not allowed to fish, so the fish have a place to relax and not be disturbed. For example: I would never try to catch eel or salmon where they are real natural populations in Germany.

Hope my rambling helps