First Amendment doesn't "expressly mention the separation of church and state."
It does not mention the exact words "separation of church and state," but it's about as expressly mentioned as you can get without bickering over exact phrasing. If the government can make no law that's "respecting" a religion, then how is there not a separation of church and state? The common meaning of "respecting" is "with reference or regard to." You know...use the basic definition of words, instead of "deconstruct[ing] every syllable of every word, to parse every phrase and determine what their interpretation of the word 'is' is." If the government can't even make a law referencing or in regard to a religion, then how could there be any juncture of church and state?
The Constitution also doesn't mention the "right to privacy" in a general sense, and in fact only expressly mentions a right to privacy in specific circumstances (right to privacy in your beliefs, to not house soldiers, to no undergo unwarranted search and seizure, and to not self-incriminate), but it exists in a more general manner than just those circumstances.
While I realize that's how the SC has typically twisted the phrase regarding the "establishment of religion" I disagree with that interpretation. The SC isn't infallible. Study history. That phrase was meant to prevent the government from creating a "Church of the US" and forcing everyone to be subject to the rules of that church as had happened when the "Church of England" had been created and used to impose the personal will of Henry VIII (who only wanted a divorce).
While you may want to attribute a specific definition to the clause based on your own reasonings, it can't exactly be said that the phrase was meant only to prevent the U.S. from establishing an official church. This can be seen in many different ways.
First, the intent of the founding fathers. James Madison, the father of both the Constitution and the First Amendment, consistently warned against any attempt to blend endorsement of Christianity into the law. Not solely the creation of an official church, but mere endorsement of a religion. This reflects the wording of the First Amendment, which references "respecting" an establishment of religion. Thomas Jefferson is the source of the phrase "separation of church and state" from a letter he wrote to the Danbury Baptist Association in 1802. The current wording of the first amendment is due to Charles Pinckney, who urged that the U.S. should pass no law on the subject of religion. Not just a law establishing a church, but any law relating to religion, which again is reflected in the wording "no law respecting an establishment of religion." George Mason argued that no religion should be "favored or established by law;" there wasn't merely a ban on the establishment of a religion, but also favoring a religion. Again, very similar to the pretty plain wording "respecting an establishment of religion." The list goes on and on, but the point is that it's pretty hard to argue that you know what the first amendment "really means" when so many of the founders who wrote the amendment directly disagree with you.
Second, other portions of the Constitution. Article VI, for example, states that Senators and Representatives should not have to pass a religious test in order to be eligible for office. Establishing a test is not the same as establishing a church, yet it's specifically prohibited in the Constitution. If the Constitution was only meant to prohibit religion being involved with the government to such an extent that only the creation of a U.S. church was banned, then why include this article? Why use a generalized phrase in the first amendment that broadly states that no law shall be made respecting an establishment of religion? Why not specifically ban the creation of a U.S. church and leave it at that? Because there was an intent to create a pretty strict separation between church and state so that the government would not be viewed as favoring one religion, and thus favoring those who practice that religion, while alienating citizens who have different religious beliefs.