We only need to look at nature to see that everything dies. It is natural, and necessary.
I am not in any way saying that people should be euthanized or disease not treated or that people not live out their natural lives. But artificial extension of life will only result in those who can afford it staying in power forever.
That's a logical fallacy. The argument you are responding to is not logical, but rhetorical (argument by analogy, based on the premise that analogous systems behave in similar ways). Logical fallacies don't apply to rhetorical arguments in a logical way.
We can all do better than pedantically pointing out logical fallacies, as no argument outside of mathematics is completely logical.
We only need to look to nature to find the immortal jellyfish.
Much of medical science is "artificial" extension of life. Natural would be you dying of Smallpox at a young age. Instead you've artificially extended your life with a vaccine.
I am not in any way saying that people should be euthanized or disease not treated or that people not live out their natural lives. But artificial extension of life will only result in those who can afford it staying in power forever.