Like you, I watched and enjoyed mountain men for long time. I grew up in the mountains, hunting and fishing, but never as remote as they are. Living off the land can be a hard life, but it is also one of total freedom. There is no job, or laws, or rules, but it comes with a lot of sacrifice. They are a rugged bunch, and I really admire how they have carved out a life in the wilderness, and have learned to depend on each other and neighbors for survival.
As to whether I think it's authentic, yes I do, because of their knowledge and challenges, and even failures at times. However, I am also mindful that there is a camera guy or crew there to film everything, so what you're seeing is probably edited and at times they may do things that just make for a good episode.
That being said though, I don't think one can dismiss the knowledge and skills Tom has in tanning hides, or his wife sewing them, or Eustice's knowledge of how to do just about anything that a backwoodsman could possibly know. They are all incredible individuals living a life that very few could tackle. Civilization and society just isn't their thing, but still, it's a very challenging life with many hardships.