This is said with an awful lot of certainty for something that's never been accomplished before. Do people and systems get stuck in their ways? Absolutely. But they also change for better and for worse. There's no guarantee/proof that living forever ends in stagnation or an unsustainable culture.
Which society of immortals are you referring to?
More seriously, it's a common theory that age-related diseases are an evolved adaptation because over time a species must form an equilibrium with its resource base.
I don't even think it is necessarily bad in toto. The other pole could look like stagnation fueled by indecisiveness and fear of possible consequences. I definitely know some individuals who spend their lives like that, and they don't seem to be happy.
But the conversation is always "immortal billionaires".
It's a totally fake problem to talk about that.
As for stagnation, there is an unspoken presumption built into your thought, namely that older rejuvenated people will gain younger bodies back, but that they will stay mentally old. I don't think this will be necessarily true. It rejuvenation works, it will have to work on the brain too, and rejuvenated people will revert to younger patterns of thinking as well. Only with more memories and experiences, which could actually improve their results. For example, they will be unlikely to join some populist cause if they burnt themselves twice already.