The government does not have some magic wand that can "bring down the cost of health care." It can buy a smaller quantity or lower quality of medical care, as other countries with government-run medical care do.
It can decide not to spend as much money on the elderly as is being spent now. That can save a lot of money-- if you think having a parent die earlier is a bargain.
The idea of a "duty to die" has been making some headway in recent years around the fringes of the left. It is perfectly consistent with the fundamental notion of the left, that decisions should be transferred from ordinary citizens to government elites.
Liberals don't have to advocate it. But, once you have bureaucrats empowered to decide what treatments you can and cannot get, they may well decide that money spent keeping some 75-year-old grandmother alive for a couple of more years could be better spent politically by enabling ten younger people to have acupuncture or visit a shrink.
Even if her children or grandchildren are willing to spend their own money to keep grandma alive, when bureaucrats control the necessary technology or medication they may decide that it is not for sale.
Those pushing for government-controlled medical care say that you can keep your doctor. But bureaucrats in Washington will decide whether what your doctor prescribes will be allowed. Talking about your doctor is another distraction from the crucial question of who will actually have the power to decide, which can be the power of life and death.