The men who wrote and ratified the First in many cases went home to states that required tithes to the established state church. This was a common practice in New England up till the 1820's. One could belong to the non-official church and send one's tithes there. But in order for these tithes to count under law, the church had to be chartered by the state.
Lawsuits were brought by non-church members seeking to avoid the tithe requirement, with the result that their tithes must go to the established church if they did not belong to any other. All citizens had to tithe to some church, somewhere, period.
Compared to today's strict "church state separation" that removes even historical recognition of religion in the culture, that was a pretty tight relationship, even in the day when Jefferson penned the famous "church state separation" quote in the letter to the Danbury Baptists.
So the question becomes "how does the 14th extend the prohibitions on Congress to other government bodies". In my humble opinion, the First is not covered by the 14th at all, because it does not enumerate rights of the people. It is *claimed* to do that, but the clear reading of it with the understanding of how governments commonly related to churches and the press at the time, it can only be a congressional restriction.
The 14th was written immediately after the civil war, and it's intent was to halt the claims of southern states that the rights contained in the BoR only related the federal government. That's a pretty silly claim, because when it says "The right of the people to keep and bear arms", the BoR isn't limited in scope in any way. That right must have related to the people living in the states, otherwise it makes no sense.
This contrasts with the First, which as I claim above is limited in scope specifically to the Congress. I trust that if the Congressional limit in the First was to be expanded to State legislatures and teachers in classrooms, the writers of the 14th would have said so. But without such modification, I think it's pretty silly that a limitation on the Congress magically limits what a public school teacher can say and do merely because the "Rights" contained in the BoR are extended to the states.
All just my opinion, of course.
I think you're generally correct here. This is the reason it took constitutional amendments to give federally guaranteed suffrage to blacks and women AFTER the Fourteenth Amendment was ratified.
Of course today, with judicial activism, the 14th is held to even prevent states from blocking the enactment of "gay rights" ordinances or having a male-only military institute.