Why would it? People always seem to assume that with extremely long lifetimes, humans would still reproduce at roughly the same age as they do now. I just don't believe that would be the case.
So many things change with lifetimes that long and it seems more likely to me that the human timeline would adjust to the new lifetime. Physically maturing into an adult may take the same amount of time, but now it's a tiny fraction of your overall life. You may spend 50 or 100 years in secondary education, then another 200 in a career at which point you should have enough money put away to live on the interest.
And it's at this point, when you're 300+ years old that you start having children. On this timescale, you'd still see more generational overlap than we have today, but no so much that overpopulation is a concern.
In the case of men, sperm quality declines with age, so a man is more or less fertile his entire life, but that may not hold true in the case of extreme longevity. If this decline in sperm-quality is due to age damage, then an age treatment may also improve sperm quality. But this is not guaranteed and may require its own specific treatment.
Women produce all the eggs they will ever produce before they are born. Menopause occurs when a woman runs out of eggs. So in the case of females, an anti-aging treatment is unlikely to extend her reproductive years beyond the current age of menopause (although it may make pregnancy safer and more viable during the latter half of her reproductive years). Like sperm, a woman's egg quality also declines with age, creating further difficulty.
So the idea that women will put off having children for a couple centuries seems to be a nonstarter lacking other advances as well, but it may be a possibility for men.
So many things change with lifetimes that long and it seems more likely to me that the human timeline would adjust to the new lifetime. Physically maturing into an adult may take the same amount of time, but now it's a tiny fraction of your overall life. You may spend 50 or 100 years in secondary education, then another 200 in a career at which point you should have enough money put away to live on the interest.
And it's at this point, when you're 300+ years old that you start having children. On this timescale, you'd still see more generational overlap than we have today, but no so much that overpopulation is a concern.