Not necessarily. Hanging around longer could provide more of a resource drain than the relative benefit, humanity aside, and older people can create limits to societal change that could be a governance beyond material aspects too.

This will sound callous, but some older people really should be allowed to die. There's definitely points of diminishing returns for all involved; I hope that under certain conditions (eg dementia) I'll be allowed to die when I'm decrepit.

yes. you need to reproduce and to ensure your offsprings reach sexual maturity. but humans are in no way special. we believe we have escaped our biology because of the cultural stratum we posses (ie exchanged memes for genes) but we are still basically animals.

Why couldn't it be true that we want to survive to ensure our grand-offspring and great-grandoffspring reach sexual maturity as well? That's a valid strategy (with tradeoffs, as there are with everything)

we definitely want to survive, even with 0 offsprings. That was not the point. The point was the natural selection does not optimize for the longest surviving humans.

Read the answers in the link I gave -- they give seemingly solid answers (I am no expert).