Most of my elderly rabbits have been euthanized due to pain and suffering. I have had one drop dead but she was just over three so I wasn't expecting problems from her. So yeah I think I'd let them die naturally if they would. So far that seems to be a very rare occurrence.
I don't plan to. My first rabbit, black Fury, is showing age at 9 years, but so what. I will care for her as long as it is feasable. Definitly would not put them down just because of old age, or getting inconvinient.
My situation is kinda special though, the line between lifestock and pets vanished long ago. No other living thing than Fury lived with me that long all the time. Herr Hase, my herd buck became my free range house bunny, and I got one of his daughters spayed as his cuddlebun, Now he's suffering from hip athrosis and needs pain medication every day.
I let them live out their lives. They are so capable creatures, actually I think they can deal with it. The evening Magda, 7yo, died I could not find her in the garden, quite unusual. Around 10pm I found her down by the creek, napping, when I woke her she went straight back to her hutch where she died that night. i still feel kinda bad to have moved her from the spot she had chosen to die. On the other hand, she wasn't alone, her daughter Chantal was with her. I guess that's one of the aspects Watership Down portraied accuratly, the way rabbits prefer to die.
I killed severly injured rabbits, that's not the same thing. Also, I'm a meat breeder anyway, so you can belive me that it's all about what's going on in our heads.
The old buck died after several days of inactivity. He mostly just laid on his stomach stretched out in a cage with 3 does. I didn't he was in any pain. He did eat and drink. If it was a person I would say he died peacefully in his sleep.