Scalia had a lot of self-contradicting points. I read all 64 pages.
It appeared he wanted to say the right thing, but was restrained by something to not "let loose the stops." The political, social and other fall-out if he had nixed the 1934 ban (just picking one) outright etc would have turned this country on its head figuratively with lawsuits, people getting out of prison etc. It just would damage the "status quo" too much for him to give an 'honest,' summary. His reasoning for upholding the 'ban' on machineguns was completely a circular argument... saying that they're illegal because they are "not in common use"... wait a second, maybe they're not common because they're illegal!
Essentially I agree with you about the National Parks, but later in his ruling he said that "sensitive places" like schools and government buildings etc could have a legit/legal ban on firearms etc. So, in one place he says something that seems to support "bear and carry" anywhere, and other places he contradicts or modifies what he says.