I know Brotherhood of Steel is a descendant of US Military, but do they still remember America? And do they ever think about restoring it? Like Enclave do?

The Brotherhood of Steel's main goal, is collect technology, figure out how it works, learn to make more of it, and prevent the people of the waste from using it incorrectly, or using it to blow up the world (again).

Personally, I don't think they give a damn about who the next leaders are, as long as they don't blow up the world.

They are decendant from an army unit that was stationed at Mariposa Military base for security. They found out what was going on there, they killed the scientists there, effectivelly rebelling. They then tried to tell the world what they had done, but the Great War happened shortly after...

So I don't understand where people get this idea that they are descendants of the United States Military. They aren't, not by a long shot.

As tybat11 explained they don't give a rats butt who is running this, just as long as they don't get their hands on the good pre-war tech and miss use it. As we can see it is no longer working out for them.

The Midwestern Brotherhood on the other hand are (could be) rebuilding, but they aren't rebuilding America. Their own nation state. But from what we can tell, they have forgotten their own history.

Lyons might also be rebuilding, but again, not America. If we are to believe him, he still feels that he is doing his orginal mission. So it's possible he could pack up and head back to California.

Just answered your own musing there. They were part of the US Military, and then went rogue, but they still are descended from it. Just because the officially cut ties from it and no longer share the ideology, it doesn't mean they didn't come from the US military.

What I get from people saying "Brotherhood of Steel is a descendant of US Military" is that they believe they are what is left of the United States Military.

They aren't. Yeah they came from a group of army guys but that's where it ends.

Well, I thought they're remnants of US Army, read something to that effect from wiki, so thanks for clarifying.

Ah, gotcha. I can see why they would think that; there's just not enough info presented to players outside of FO1/FO2 that says that the BoS doesn't think of themselves as US Military remnants. At least nothing directly stated.