I practice a style of Bagua called shishi baguazhang, and it has been thought to be extinct, so I was informed for over a 100 years. There are many styles which have gone extinct, but there are quite a few I imagine are held to be extinct, and there are still a practitioner or two who is handing down the training, master to student, though one at a time, and as quietly as possible. This is how I learned my art, and my master from his before him. We go far back as well, and keep the history well guarded. Past the 5th master back, for example, it gets murky as to where the art really developed, stylistically for the branch I practice.
When my master passed four years ago, as far as I know, I am the last practitioner of the style. How does one verify that? They can't really, and it can be annoying, but you work with what you get.
There are lots of claims to learning 'secretive' styles with little evidence, but I think this stems from assigning what almost feels like a value system to the arts. I feel that martial arts are expression of thought through physical means, and as such, each style is like a language in being representative of thought- and of a certain kind of thought. Some forms of logic work better in certain situations, and likewise, so too do certain arts, though none are superior.
I was not the only student my sifu trained, I am merely the only one he trained after WW2, and that he knew of no surviving students. When you become the successor of anything, you find yourself, I think, seeking students who are of quality. It would pain me to teach shishi bagua to a 9 year old who quit 3 months later, when she was just dipping her toe in. I would not teach it to a person like that, and I do not believe my master would either. It is not that the art is kept from people, it is that it is reserved until ready to be expanded. I think of Wing Chun, which was passed quietly through several families for many decades before several key figures, across two generations were able to expand it to its reknown today, and even foster subsequent evolutionary, new styles.
I imagine, like language, there are some which are on the verge of true extinction. And I imagine many of the last practitioner of anything have found themself wondering between whether they really are the last one, and its implications, or if they are just separated from the others like them. It's a difficult fjord for most to cross I believe.
In the end, every art is its own, and the practitioner. One could argue every style is extinct, and no true standard really exists anymore. Every style changes with each generation. No matter how many safeties are placed in an orginization to mitigate this. I think of kendo, and its modern rigidness, and how in the 30s and 40s and 50s it gradually lost the throws and its kicks. Of how Tae Kwon Do arguable didn't exist until this century (TKD <3). I imagine when I teach shishi, it will be different, in grades, than what my master taught me. But I am not him, and don't believe I ever will.
I ramble, but my point is this- an art is an art for the value it holds in what it can communicate across divisions. I have found no art perfectly capable of that, yet.