How great is the utility of "saving" endangered languages?

Being bilingual has been linked with a lot of benefits

Being bilingual is AFAIK a strong signal of cognitive competence: given a choice between 2 applicants for a cognitively-demanding job, one bilingual and one not bilingual, I would heavy favor the bilingual one. But that does not mean that investing effort in making a person bilingual increases the person's cognitive competence to any significant degree.

One thing we don't need studies or complex arguments for is the fact that it takes a lot of study and practice to learn a second language -- time and mental energy that can be used to learn other things. Our society has accumulated an impressive store of potent knowledge, knowledge that takes a long time for people to acquire, but which clearly improves their lives and their ability to contribute to society. I'm very skeptical that the benefits of spending an hour learning a second language outweigh the benefits of spending an hour learning, e.g., history, geography, chemistry, physics, statistics, computer programming, practical human physiology, cooking, sewing, woodworking, accounting or the basics of public speaking or performing in front of an audience.

I'm anticipating that you will reply here that there is more to culture than knowledge that has obvious practical benefits. And my reply to that is that I don't see why an hour spent on second-language learning would outweigh the benefits of an hour spent watching Buffy the Vampire Slayer or Desperate Housewives or browsing https://tvtropes.org/. Those 3 things are products of the dominant culture, and I suspect that most of the effort to save endangered languages stems from a perceived need to fight the dominant culture in any way possible (but I don't perceive any need to fight against the dominant culture).

Culture is important, I absolutely agree. This is not the same as claiming that there is a way to save the culture by way of saving a language, in a way that preserves the important bits long term. I think that most of the value is reflected in the language but mostly contained in the living traditions and experiences and practices of the people who generated the language. Preserving the culture is a whole lot more complicated than preserving the language, and we don't really know how to do it effectively.

So what's the long term goal? Do you raise native speakers living mostly in the culture from which the language originated, partly isolated from the outside world so they think in the smaller culture's language and memeplexes? Raise bilingual speakers who mostly think and speak in the dominant culture's language and memeplexes but abstractly know bits of their ancestors' cultures? Train non-native speakers in an attempt to document stories and practices that are no longer a living tradition? How much value do the different options preserve, and for whom? Who should be responsible for making and paying for the decision to try one method or another?

Consider ancient Greece. It's a pretty good example, since without the language and a comprehensive knowledge of the culture there's a huge range of meaning you lose out on when reading (or hearing) the surviving works of literature/philosophy/history. Millions of person-years have been spent trying to improve and advance our understanding in this field. In the Middle Ages, this was really important: reintroducing Aristotle to Europe (via the Middle East) was a huge gain in value. In the modern day, it's really cool and helps us understand history but is otherwise mostly a curiosity; the kind of thing you put a little effort into for the really impactful dying or dead languages. We don't even attempt that for most of the other languages that existed in Britain in Chaucer's time, because the value just isn't high enough to justify the level of effort required.

Now, if there's a way to sustain a language and culture long enough that it would be here until we got a future technology to preserve it better long-term (either AI or biological immortality of brain-machine interfaces for rapid learning and teaching) then that would be different. If you only need to attempt preservation for another generation or two, after which you get more preservation very cheaply, then that's a great deal.

Being bilingual has been linked with a lot of benefits

It is plausible that bilingualism somehow exercises the brain, but it seems to me that a much stronger case can be made in the opposite direction -- smart people are more likely to successfully learn multiple languages.

I agree that it is good for children to grow up in a bilingual environment, because they get the other language for very little opportunity cost, if the other language is naturally around them.

Preservation of endangered languages involves raising children bilingually in the majority and endangered language...

Generally speaking, endangered languages are from a cultural minority and members of that minority culture enjoy being able to speak that language.

If the minority cultures can fix the problem themselves by teaching their children, great! Far be it from me to stop them from that. And of course the dominant cultures should not actively oppress minority languages.

But when outsiders are expected to put in extra effort to preserve minority languages - that is when I balk.

Imagine if english went extinct. In a sense, we'd lose Chaucer, Shakespeare, Austen, Steinbeck. ... These "endangered" languages had culture too - songs and stories, maybe books and plays. That's important.

Important, sure. But other things are much more important, such as eradicating diseases and getting people basic education and preserving the environment.

If I had the choice between saving just one (decent quality) human life and keeping an endangered language alive for another generation, I would sacrifice the language to save the human.

Close, but not quite.  There well may be colors and gradients, but we're all very nearsighted and can't tell what anyone else is seeing, and there's no photography or ability to share the experiences and intuitions. There is no authority to tell us, no way to be sure that what we think is important is what matters to somebody else.