Why would you not want to stop people from dying?
If you end that cycle of renewal, the rate at which our society reacts to change and adopts new ideas would significantly slow.
That's a problem regardless of how well the society aligns with my ideals.
Personally, I think one of the biggest challenges to transitioning to a low-birthrate, ultra-long-lived society is that a constant influx of young blood seems to be a necessary engine for new ideas and progress. There's a combination of not knowing better[1] and not having as much to lose that makes young people more willing to take big risks and push for more egalitarian societies over time. Left to their own devices, accumulation of wealth and status tend to ossify people's belief systems and reduce their incentives to throw everything out and start over.
[1] by this I mean not having the same accumulated beliefs about what is best. Sometimes those beliefs turn out to be unfounded, and you discover that by challenging them.
To quote Max Planck, "Science progresses one funeral at a time." That's as true of anything as it is of science.
Actually, the opposite is true to a degree. Limited resources increases the birth rate.
The only way around this is advances in technology that improve food production/distribution, but if this were to occur, then population would no longer outstrip resources and a new balance would be reached.
Last time I checked space is pretty huge, we could move there.
And if you live long enough, you might not need a biological body to survive.
For example: "Internet freedom around the world has declined for the fifth consecutive year, with more governments censoring information of public interest and placing greater demands on the private sector to take down offending content. "
if done incorrectly, this might end up in quite a disaster
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.