Many people have very painful and terminal debilitating diseases that means they cannot move, talk, eat or wash themselves properly or not at all. A lot of the time they need assistance. I myself have never had this problem, but just to imagine being sick every day of my life, in excruciating pain and not being able to go to the toilet myself is just embarrassing to think about. Many terminal illnesses take way dignity in many ways, and having the right to die can restore some dignity to those who feel like when they lost the function of their bodies or their health, they no longer had a choice.
If someone is old and has a terminal illness then euthanasia is ethical. Often, terminal illnesses cause a lot of pain and hardships, not only to patients, but also to their families. Sometimes, the most humane thing is to let a person die with dignity and peace rather than making them suffer longer.
Yes, I think really sick elderly people DO have the right to die. It is their choice, and really if the quality of their lives are awful then they should be able to make a conscioune descions, they do anyways when the family make them a DNR do not resistuate.