Does life extension (without other technological progress to make the world in general safer) lead to more cautious life styles? The longer the expected years left, the more value there is in just staying alive compared to taking risks. Since death would mean missing out on all the positive experiences for the rest of one's life, I think an expected value calculation would show that even a small risk is not worth taking. Does this mean all risks that don't get magically fixed due to life extension (for example, activities like riding a motorcycle or driving on the highway seem risky even if we have life extension technology) are not worth taking? (There is the obvious exception where if one knows when one is going to die, then one can take more risks just like in a pre-life extension world as one reaches the end of one's life.)
I haven't thought about this much, and wouldn't be surprised if I am making a silly error (in which case, I would appreciate having it pointed out to me!).
There's 2 factors here.
Suppose there's a life extension treatment that resets someone to age 20. It's readily available to most first world residents, with the usual methods of rationing. (wait lists for years in European countries, the usual insurance scam in the USA)
A rational human would yes, buy space in a bunker and do all of their work remotely. There would be many variations of commercially available bunkers and security products, and the recent pandemic has showed that many high value jobs can be worked remotely.
However, the life extension treatment doesn't change the 'human firmware'. Novel experiences and mates will still remain highly pleasurable. Staying in the bunker and experiencing life via screens will likely cause various problems, ameliorated to some degree with artificial means. (vr headsets, etc)
So there will be flocks of humans who keep taking risks, and they will do the majority of the dying. I think I read the average lifespan would still be about 3000 years, which seems like a large improvement over the present situation.
In addition, this would probably be just a temporary state of affairs. ('a dreary few centuries') Neural backups, remote bodies, deep dive VR - there are many technologies that would make it practical to go out in the world safely. And a survival advantage for those humans who have the neurological traits to be able to survive the bunker years.
But, yes, I also think that society would slowly push for cleaning up many of the risks we consider 'acceptable' now. Cars, guns that are not smart and can be fired accidentally, air pollution, electrical wiring and gas plumbing - we have a ton of infrastructure and devices where the risk is small...over short present human lifespans. Everything would need to be a lot safer if we had expectations of thousands of years otherwise.