Ending aging will not reverse this trend. Many people only have kids due to age constraints and its effect on fertility and the health of offspring, ie. much harder to get pregnant over 35, and chance of birth defects or other problematic genetic problems increases dramatically, even for men.

This will no longer be the case, so there would be no rush for either gender to have kids. You can focus on your career for the first hundred years, and then have kids. At worst, there would be a minor uptick in population while people get accustomed to the new reality.

There's a really simple solution to this: Have a cultural policy that what's normal/acceptable is to delay having children for quite a while, and then to have only one child per couple.

One child per couple means that the total population caps out at double the current population (a population of 32 would have 16 children, who'd have 8, who'd have 4, who'd 2, then 1 person who's SOL - 32+16+8+4+2+1=63). A single order of magnitude is an acceptable loss, and everyone gets a child of their own.

Note: Not as in "literally force people to have only one child", just make it frowned upon like smoking generally is - most people won't care enough to defy it, and in the long term it should be enough. Although IMO families like the Duggars[1] shouldn't be provided immortality unless they start using birth control.

[1] https://en.wikipedia.org/wiki/19_Kids_and_Counting

Ending aging won't stop accidents, homicide, suicide or (probably) even most modern physical failures like diabetes and heart attacks.

It might fix the latter. The mechanisms involved in heart disease and diabetes are related to general aging as well.

Accidents and suicide could be handled too. Former by outlawing machines of murder (cars) and spare or artificial organs, latter by improving our understanding of neuroscience and psychology.

yes and no. Aging research wont, but that does not mean other science stops. Traffic accidents are at all time lows (in comparison to the amount of traffic) and will continue to do so (via automated driving, imidiate ambulance dispatching, etc.), homicide as well (maybe not in US but certainly in europe), and depression is under active research like being partly caused by the microbiome etc. Nothing is endless but getting 600 years old would still change a lot.