Why do doctors want to keep a patient alive that doesn't want to live?

They can't: If the patient signs a dnr and/or has advanced directives, they can not keep them alive against their will. This has to be done while they are of sound mind a body.

It's complex: Under the law, a person who is seriously sick is entitled to refuse treatment and to demand a comfortable end of life. Obviously, we do not allow depressed people to destroy themselves. Some (by no means all) physicians resist this for their own reasons, just as a mechanic might resist a person choosing not to fix a car that could be fixed easily, for a while. And we're all prisoners or economics.