Hi, welcome to Yorkie Talk
I don't have first hand knowledge, but as far as I know, when you're talking about a good rescue, they usually do everything possible before resorting to euthanasia.
We all know there are times when a dog's health is past the point of being able to do anything to give that dog any kind of a quality of life no matter what resources are available, so I'd imagine rescues only make that desision at the recommendation of a vet who understands that the condition is beyond help.
As far as mental illness, working as a vet tech, I have seen 2 cases where there was just something wrong psychologically (possibly from bad breeding) to the point that the dog was a danger to others no matter what training method was used (one Boston Terrier comes to mind that was actually tested for rabies after euthenasia he was so crazy, owners tried EVERYTHING, the dog was just practically rabid (test was negative) and was biting people left and right for no reason at all even after extensive training).
Even though it's sad, I think rescues have to do what's best in the long run, just like owners do. I'm sure they do everything possible first though (some humane societies and SPCAs are a different story though, they don't have the same finances and resources and the volume of homeless animals is too large for them to go to help each one). They have to have that "disclaimer" in there just in case, although I doubt it's used except for in extreme or terminal cases.