Liam,
That video is actually really interesting in the sense that shes playing on a Jiuta instrument (chuzao) since Jiuta is her main style, but she is using Nagauta bachi and koma. I am pretty sure the piece is fairly contemporary, so she must’ve changed the koma and bachi to fit the style of the music.
Tokiwazu (Tokiwazu Mojibei—Super jealous you got to see him perform!!!) and Kiyomoto (the two are sibling styles) both use shamisen with the hosozao style hatomune. I dont really know the complete reason for it, but these styles do indeed use a chuzao size neck, larger bachi and larger koma. I believe that because of the use of a larger koma, the strings are farther apart and require the larger surface area of the chuzao neck.
My teacher has told me (and I have found to be true) that nowadays even Nagauta shamisen are not really hosozao. If you look at old hosozao shamisen, the necks are a good bit thinner. I think its because the other styles were getting envious of Tsugaru shamisen and its popularity, so everyone wanted their shamisen to be bigger. (Freud must be so proud).
Cana,
Your shami doesnt have the nagauta hatomune. Check out my profile picture and compare it to your shami. Chuzao hatomune “swoop” downwards, but not until it gets fairly close to the dou. Hosozao (and Kiyomoto and Tokiwazu shami’s) have a gradual slide.
Here is a video of a minyou teacher playing for her students… Maybe you’ve seen it? There are more minyou videos on that users channel as well 
There are probably a lot of minyou teachers in Japan that have websites, so maybe google the kanji and see what comes up?
Also if you are interested: there are scores for minyou if you want to learn folk songs. The most popular are the books in red and the books in orange. You can get them from Kameya, but I’m sure Kyle can get them too!
I hope you can find more information on minyou soon! 
Joe