Hello Zack, welcome to Bachido.
First, let me start by saying that shamisen subtypes are just different takes on the same instrument. What you can do on one, you can do on the other. The biggest change between the subtypes is what gets emphasized.
Even a nagauta when played appropriately, can and will have a percussive character to it.
Take a listen to this player
The strings are definitely at the front and center - but again and again we here the percussive snap of bachi on kawa as punctuation and contrast.
Or if you prefer, here is Kyle Abbott playing a Tsugaru piece on nagauta shamisen - the one available on the store in fact!
In both cases the snap is a little milder than your video - but it remains present. Proper form means getting that balance of percussion, string, and characteristic sawari buzz.
But why is it milder?
To the best of my knowledge, the two largest contributors are play style and the physical limitations of the instrument.
A nagauta, even if skinned with dog or the like, is going to have a thinner, more delicate skin and a smaller resonator compared to the heavy and heft tsugaru.
A smaller resonator means less sound, and a thinner skin means an easier time picking up smaller vibrations (say, off the strings). A thinner skin, as Kyle implies in the video, also means playing a little more gently.