I doubt we'd face overpopulation… at least for a few millennia, but on timescales like that you need to ask about Dyson swarms etc. anyway, not just this planet.

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.

arguably, we're facing overpopulation right now

We don't have overpopulation. Statistics show that in the current mode of food production, we can support a few multiples of the population we have today. But the economic system does not even permit supporting what we have, because it needs to make food scarce and extract maximum price for it to profit. That's why millions of tons of food rot in warehouses somewhere around the world to not disturb the food prices in the market while people starve somewhere else.

In such a system, you can never solve overpopulation - the less people there are, the more system will try to make things more scarce to extract more profit from those people. A system built to be inefficient (from our perspective) so that it can be efficent from the profiteers' perspective