The freezer is what I've been advised to do, also by someone in the pet store trade, who said that's what they did with the finches & parakeets that were ill. Put them in a paper lunch bag & set in the freezer. They go to sleep & then die. I have done that with little chicks who were sick beyond hope, and seemed to be suffering. But I think if it was a chick with some awfully painful injury I'd give it a quicker end & give it a snip or chop through the neck.
You're right, this is a very sad subject to consider. But it's essential that we know how to care for the birds entrusted to us, and that includes knowing how to best put an end to their suffering when we must.
A friend of mine had a very sick hen and decided to put her out of her misery by setting her in the freezer overnight. The next morning they opened the freezer and the hen walked out, fully recovered! They changed her name to "Miracle" after that.
Have you heard the joke about the profane parrot? His owner had tried everything to get this parrot to stop saying bad words and finally, at his wit's end, decided to shut him in the freezer. The next day the parrot begged his owner to let him out and promised to never swear again. After his owner agreed, the parrot said "I know I've been a very bad bird with my profanity, but tell me, what in the world did that turkey in there ever do?"