My contention with the idea of cultural stagnation with immortal populations is birth. Idealistically, an immortal population would always grow should the death rate fall that much, meaning there will always be a considered youth. I'd imagine generational identity will be big as it is now. There is also the matter that much of modern culture is cyclical and is considered under more factors than simply new generations. There's geography, group identity, events and trends that define cultures.
If anything, I'd be more interested to see the sociological implications of an immortal society. I was going to write a book about it so long ago about a polynesian island that discovers the fountain of youth and so explore the sociology of such a society.
We have institutions and structures today, and as long as there's a constant feedback loop of what works and what doesn't within that system, then it'll be constantly self-improving. But the problem with immortality is that of pervasive personalities. People imagine that personalities change, but one example where personalities can't change is politics. Political personalities and their existence relies solely on their representation of constant ideals. Should they change their personalities, their opinions, and their beliefs, they can be called untrustworthy or a flip-flopper. This nature of a constant, unchanging political personality invades all aspects of institutional structures. Imagine the CEO that rallies under one certain program, a university bureaucrat whose identity is tied with a theory, and of course a political figure of a particular ideology.
But this issue can thankfully be easily resolved with limited tenures like term limits. Except that's in an ideal world. Russia and China had short term limits. The US used to not have term limits. And what of modern issues with regards to generational stagnancy? What if the trend of a falling birth rate continues, and what we're left with are younger generations who not being able to feel represented by older generations in the upper levels of government, and so cease to participate in democracies, leaving older generations far more over-represented? That's what's happening now in many western democracies.
I think the fault that people take when they think of immortality is the fact that since we're immortal, we have an infinite time to change when that's not the case. Psychologically, we are limited with our biases and our constraints. When we take to a certain way of thinking either due to genetic, developmental, and social influences, people are very very good at sticking with it to the point that I'd imagine people could remain the same for centuries until their accidental deaths. More-so when environments people enjoy are likely to remain within their lives for a longer time as they keep comfort towards them.
This is interesting to explore and it'd be interesting to apply motivation research to immortality. I need to fucking study what am I doing.
EDIT: I forgot the force of competition, which is very dynamic and may very well reduce this. But again ideal world.