First, eternal youth isn't a cure for all illnesses. Second, even if all illnesses were also cured and the only causes of death were injuries, malicious or accidental, the average human lifetime would be something like 1000 years (at current rates). Third, as nations get richer, the reproduction rate goes down, and in many cases is currently below replacement.
I also don't expect it to only be available to the rich. The first 5-20 years perhaps, but in my lifetime full genome sequencing has gone from impossible at any price to a few hundred USD. I expect similar for anti-aging. Even if the stuff is patented and restricted, I can say similar about heroin and cocaine, and yet the black market for those exists.
