Actually, I've never heard the word "euthanasia" used in casual conversation. When someone is going through that, they go from the hospital to a hospice - and when you talk about the person, you just say something like "he's in a hospice" and everyone knows what that means. So all this talk about the terminology seems a bit odd and technical - and interesting, perhaps because those terms don't get used often in real life.
I don't know which is harder for the family ... the idea of removing life support, like a feeding tube ... or hospice. With the first, death comes much quicker - but the family had to make the decision. And that's got to create some guilt to live with. With a hospice, the family's life goes into limbo, waiting for the loved one to die. (I know this - I've been through it.) And every day the loved one doesn't die is another day of mounting stress, which quickly reaches epic proportions. Even though the average hospice stay is "only" around 2 weeks, it seems like an eternity.
But the idea of the hospice is a natural death, controlled to be as comfortable as possible. It's the opposite of guilt. Given the inevitable, you come out thinking "at least we made the end as good as we could."
Meh. Let's just solve all the problems by building some of those suicide booths from Futurama.