You have pretty much summarized everything. If languages were such a valuable thing, why not constructing new languages? why not promoting the elvish and the klingon? It's not about the value of languages.
If it was about musicality or knowledge in it, they could be recorded, written, safely stored and kept there. There is no reason to keep them "alive". A language is not a living being, it's information, it can be stored and retrieved. We may not be able to clone a dinosaur, but we sure can teach a language to a baby (as growing into a child) if in the future we want to have a native speaker of that language (which would be considered cruel an unethical anyway). So it's quite dubious how ethical is artificially maintaining "alive" these endangered languages, as compared to letting them die (and maybe trying to resurrect them later, if it turns out that is good in any sense).
Some people say languages influence the way people think. Well, putting a name to something helps to conceptualize about that thing, we could consider granularity, etc. But this is a dangerous idea that finds few people to defend it, for one single reason. If languages are such an important tool, they can be better or worse for some specific tasks, and definitively the union of all of them would contain all the information and be the most useful tool. This leads to a partially ordered set (a lattice) where some languages may be better than others. E.g. the piranha tribe does not have numbers, they don't need them, this doesn't mean their language is better, you take English, Chinese, German or Spanish, remove the numbers and you don't get a better language, it's plain worse. So this leads to make the definite language and not speak any other, and people don't like this. It's not about utility or epistemic value (which would be useful).
On the contrary languages have to be considered as "invaluable", having an infinite value. Why? because otherwise they could be on a scale and something could be more valuable than that. They don't have more value than as a mean, for communication and (through communication) sharing ideas about which we can think later.
But they are considered as art, as part of the identity of the people and some other things that only have value due to a convention in their value, not any pragmatical purpose. An end on their own and not a mean, mostly to hide their own value.
The value, the purpose for which they are used splits into several.
- divide the people and control them. E.g. immigration, misinformation.
- nationalistic fallacies, as in fascism.
- caring about the people and their culture (by dividing them). Politicians like to show themselves as defenders of the languages and the people. This provides votes while doing something that makes people less empowered and more manipulable. Win-win.
This is mostly why klingon is not an option, which people is that? What culture is that?
Culture is again a way to control people, sets of individuals are homogenized into a mass of people, a flock to control, and that is much easier than controlling chaotic and unpredictable individuals.
In short is a soft-fascism, a way to divide and control people, a way to give them something that they want (or that they have been convinced to want) and look like a good guy while actually harming these people, that in the best case are human Guinea pigs (artificially keeping the language alive) and in the worst case are sheep in a flock (controlled by the shepherd).
You will say I'm an extremist because of the word "harm". Learning a language that is endangered severely limits the access to information and the possibilities of personal growth, so that's harm. You will say they can learn that language and then some other language as well. Yes, sure, but instead of those two languages they could learn English and Chinese, or Spanish, or some widely used language that is going to be more useful for them, for communicating information, for accessing information and for enriching their understanding of the world.
While actively acting against those languages may not be ethical (because that is usually done by actively acting against people who speak those languages), artificial preservation of those languages is equally unethical. Any intervention, on one direction or another is unethical, for the reasons exposed and some others that are more blatantly evident. Yet politicians and other people have manipulated many people to think in a different way, for reasons they can't really explain most of the times. I don't expect this answer to be greatly appreciated due to this. Blame democracy and lack of critical thinking.