The adage is that if we didn’t die of age we would simply not cross the street, the odds may look slim in our short lifespan, but if you extend those to a thousand years getting hit becomes a facto of when, not if

Familiarity breeds complacency. The longer people do something without having an accident, the sloppier they get.

With current population curve (and assumption of yours) states would quickly need to add mandatory accidents due to exploding population.

But imagine skills of those bright people who pushed boundaries of mankind into better places, if they could contribute for say 500 years. Like in Tolkien's world elves attained great mastery ie in smithing over millennia of experience.

In any case it would be horrible for mankind, because there is no shortage of Putins and Stalins in general population and they would burn whole world to the ground without blinking an eye to get closer to immortality. No, thank you, death is a great thing for us even if individually most of us are shit scared of it, and various religions trying to tackle this certainly don't help as much as people would like.