Why not go to the movies? Without a strong economic incentive to learn a language, any recreation is just is good. Presumably people are rational and will choose recreation that suits them. If learning rare languages is recreation, then governments don’t have as strong of a policy guidance for funding the project.
Why not learn Spanish or French (or some other potentially commercially valuable language)? These mega-languages are studied because they are the lingua franca (i.e. the most common, widespread language) for some chunk of the world, often for business reasons. This is really the same issue as going to the movies instead of studying a language, except now we assume languages are commercially valuable. Again, we can only presume that people are somewhat rational and can make choices about which and how many lingua francas they need to learn to get their job done. In the US, one can get by just fine on just English, since at the moment it is also a global lingua franca. The existence of lingua francas with well distributed learning materials may be a good thing, for the student of a dead language, because they can get a chance to learn how to learn a language using something easy like Spanish or French, and then move on to something more difficult. And by difficult, I mean, difficult for lack of resources, not merely difficult to pronounce, spell or write correctly.
Why lot learn some still spoken language? If materials are available, then this might actually be practical. Now for choosing, which one? If materials aren’t available, current speakers are far away or mostly extremely elderly, then it’s a hard sell. So in the US context, Navajo and Cree would be a better choice than any dead language. In away, the prospective student of a dead language faces the same choice as a person who could study Finnish or Chinese. There are a billion Chinese speakers, but if you live in Stockholm, you’ll probably get more opportunities to use Finnish. If there are more people nearby interested in learning a dead language, that trumps materials and a large speaker base that may live, far, far away.
Why not learn a well attested dead language? Nattick is well attested, Powhatan not so much, Nanticoke not hardly documented at all. You can read something authentic in Nattick written by fluent speakers, but not in Powhatan.
Why learn a poorly attested dead language? Sentimental reasons, for the challenge, and because one will be able to see a language in the process of being created.
Why learn *that* poorly attested dead language? If you pick a dead language with descendants who still exist, you will need to decide if you want to involve them or not. Just like with conlangs created by a single person, sometimes people get defensive about their family or town’s dead language– e.g. it’s dead and it should stay that way, or it’s dead and its mine to revive or not, etc. (I’m not sure I buy into all these potential objections, just saying someone might make some objection you may or may not want to deal with). I think it makes some sense to study a poorly attested dead language if you happen to live near where it used to be spoken. On the other hand, if you are just looking for materials for creating a conlang, Beothuck would be an interesting example of a language that could be used for inspiration or material for a new language, without much likely controversy because the speakers and all of their descendants are extinct.
And of course, the unattested dead language, can only be a pure fabrication, especially if you are reaching for proto-human (spoken in Africa 80,000 to 40,000 years ago), or one of the many languages that died out long before anyone knew how to write or thought of writing down reference grammars and dictionaries of the last speakers.