Overpopulation right now is not caused by people getting too old, it's caused by people having too many babies. It is also not a problem in countries where anti-aging treatments are likely to be available first and even less of a problem in the rich population that will be able to afford them at first.
On the other hand, the promise of remaining able bodied until you die from an accident might reduce the number of children because you don't have to have children that take care of you when you're too old to take care of yourself.
I propose we worry about that when it's actually a problem. We don't know how people behave when they can keep youthful bodies for decades longer than now. Maybe we're lucky and they just don't want more children.
It will likely also take a generation or two until our anti-aging technology actually works and is cheap enough to have a meaningful impact on the planetwide population.
Exactly, do people really think that anti-aging technology is on the verge of becoming available to people who currently have trouble getting clean drinking water or who are dying because they can't get a few dollars worth of anti-malaria drugs or antibiotics?
A lot of the world can barely afford dirt cheap medicine from decades ago. They aren't going to be able to afford state-of-the-art anti-aging treatments for a long time.
On the other hand, the promise of remaining able bodied until you die from an accident might reduce the number of children because you don't have to have children that take care of you when you're too old to take care of yourself.