I'm far less certain of that - look how many people take life-extending drugs now, so that their failing heart/liver/kidneys/whatever won't kill them "before their time"
What makes you think a drug that benefits basically every aspect of your health, plus makes you look and feel younger, would be any less appealing? Granted, after a few decades of failing health perhaps the allure of several more would wear thin. Then again, our culture is rather obsessed with putting off dying as long as possible, at almost any price, so a relatively cheap and powerful aid to that goal would probably find a huge market, even if it did also accelerate the return to a more sane relationship with death.
Even then though, I would not be surprised if death from "old age" remained extremely uncommon, with people instead continuing to take the anti-agapics as long as possible, and then choosing a more pleasant way to go when they decide their time is up. I mean it's basically a choice between decaying as slow as possible and then taking a leap of faith when it's no longer worth it, versus willfully letting the misery of decay accelerate until it kills you.