It's never easy. Having grown up around a farm, one thing I learned is that the pause one takes about butchering an animal intended to be meat is a sign of respect for that animal. And that a person takes that pause speaks well of their character.
The same holds true when an animal never intended for the table has to be put down. Over the last very few years I've had to put down a promising young horse that contracted an encephalitis, and our fine and faithful dog, Miss Maya, whose liver gave out without notice. Both painful decisions and events, but necessary.
That's what you've experienced. Any comfort with the decision or the deed is to be found in knowing that you did what was right for your bird.