And people don't usually die of "old age" but of things related to it. When the body gets older, the immune system gets weaker, your bones and muscles decay, your brain gets messed up, and a lot of deaths are in reality just a mix of a bunch of factors that just result in the body kind of shutting down. None of that will happen anymore.
And there's a feedback loop here too, because you a) have less life left to live and b) is generally weaker you get less treatment. A relative of mine is dying from cancer and it's low intensity life-prolonging treatment. If he was 20 years younger, they'd put him on high intensity drugs that would keep the cancer suppressed much, much longer. If he was 50 years younger, they'd probably try a full bone marrow transplant which is a massive procedure that is not only ridiculously expensive but likely to kill the old by itself. So being forever young wouldn't just avoid age-related diseases, it'd open up far stronger treatments as well.