The philosophical question is not if the body is alive, but whether it is a
legal living person with rights.
There is no such thing as a simple “breathing tube”. Or if there is, it
isn’t here. Since at the other end of such a tube is a monstrously
complicated ventilator, since this girl does not breathe for herself.
A headless body, very carefully prepared, could in theory continue to have a
heartbeat and metabolism. Even incubate a fetus to maturity. Is this what
you consider a “person”? If we separated head and body and kept them both
alive (as we very have the technology to do) do you think that now we have
two people? Or does that apply only if we lose the head? Or it dies?
The brain of a brain-dead person is not being perfused with blood (you can
even do a perfusion scan to see this). It eventually fills the skull with
liquefied decomposed brain, like a milkshake. If you want to keep such
mush-filled skull lumps of tissue alive on a ventilator indefinitely, then
YOU pay for it. Or let the family do it if they have resources. Which, of
course, at costs of $20,000 a week at least, they do not.