To quote Max Planck, "Science progresses one funeral at a time." That's as true of anything as it is of science.
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.
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.