The 1st Amendment reads:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof (emphasis added)
The first part prohibits the state from using its operations (like state-run schools) as a means of promulgating religion. And yes, that generally includes singing explicitly religious songs in official school events.
Further, we have carved out ways for individuals to exercise their religious beliefs within these institutions. E.g. students are allowed to pray in school, including in the company of faculty and staff, as long as they do so in private. They cannot do so in connection with official events, such as leading a prayer during graduation or before a sporting event.
What does the second part mean? It means the state cannot stop you from expressing your religious views -- in the privacy of your own home, or in the public square. It does not license the state to compel students to participate in or be exposed to religious expressions.
Along those lines, there is not, nor should there be, an exception because "students want it." (Santa Fe Independent School District v. Jane Doe) It still results in the idea that the state is sanctioning a belief system, and that students are compelled to participate. And if the students want to sing Christian songs, nothing whatsoever stops them from doing it on their own time.
I did not say that the 1st Amendment took effect before it was ratified. My point is that there is nothing in either the history or theology of Christianity which entails tolerance for other religions.
It is also false to say that the first colonies were formed out of a desire for religious freedom. The Virginia colonies were purely a for-profit venture; the Puritans had no interest in religious liberties, and claims that they fed persecution in England are vastly overstated. It was only the most extreme Puritans, who wanted to leave the Church of England, who were persecuted in any way; most Puritans were not, and considered themselves as Anglican reformers. Rather, the Puritans came to the US because they wanted to create their own utopian Christian society at a low point in their political influence. The vast majority of Puritans stayed in England, and slowly gained in political power. And obviously, the Puritans were incredibly intolerant of any other religious groups, until conditions forced them to change.
I.e. the motivations to colonize the New World had nothing to do with the ideas which resulted in the passage of the 1st Amendment.
The idea that "political correctness is the worst!" is as ignorant as it is repugnant. What has it done -- made some privileged people feel bad about saying certain words? Awwww. How awful for you all!!! Meanwhile, racial intolerance has led to all sorts of violence, ranging from assaults to lynchings to discrimination to segregation and more. We see similar outcomes due to the intolerance of LGBT individuals. Sectarian and religious intolerance has been intertwined with or causative in centuries of persecution, oppression, torture (think Inquisition) and warfare (ranging from the Wars of the Reformation era, to Iraq today); and of course, anti-semitism, which resulted in the slaughter of millions of Jews during WWII.
When President Jerry Brown Orders the Suede-Denim Police compel you to jog for the Master Race, on pain of putting your uncool niece into an organic gas chamber, let me know.**
** 10 points to anyone who gets the reference