[Spats McGee]

Start by seeing Spats McGee’s Federal Constitutional Primer

In particular, check out the First Amendment:

Originally Posted by Founding Fathers

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances. U.S. Const, Amend. I

It doesn't say "Congress shall not pass any law that affects any religious institution in any way." It says "shall pass no law respecting the establishment of religion." A religious symbol is particular to one given religion, and displaying one on gov't property is akin to endorsing that religion. Prohibiting all religions from allowing concealed carry (or open carry) on their property does not advance one religion over the others.

Comparing them to all other private property is an apples-to-oranges comparison. Bear in mind that I, personally, oppose laws that do prohibit concealed carry on any property, but it's not a violation of the doctrine of separation of church and state..

ETA: I just saw the following edit to your post:

Taking it a step further and playing the "what if" game, what if a religion actually does require the carrying of arms in worship? How could those prohibitions then not violate the 1st Amendment as they would directly impact free exercise?

That is precisely what makes this case interesting. If a religion actually did have "carry a weapon" as one its bona fide tenets, then there might be a valid A1 argument. However, I'm unaware of any religion that specifically requires the carrying of a firearm.