Jefferson wrote: . . . legislature should "make no law respecting an establishment of religion, or prohibiting the free exercise thereof," thus building a wall of separation between church and State.
Your conclusion: All the Supreme Court did in 1964 was to not allow captive or forced prayer in a public institution. It said absolutely nothing about voluntary, student-led prayer.
Had this been all the 1964 Supreme Court did, and intended to do, their decision (and subsequent decisions based on the 1964 ruling) would not have paved the way for the systematic stripping of God from everything academic (i.e., Pledge of Allegiance because it says "One Nation Under God", or public prayer at graduation services (regardless of whether led by pastor/rabbi/priest) and political (removal of the Ten Commandmants all over the country in front of city halls; bans on nativity scenes, Christmas trees, or Menorrahs), for example.
One also needs to look back to the Founding Fathers and their culture. Had they wanted there to be a complete removal of God from society (i.e., government) so as to maintain this "separation of church and state," they would have ensured they did so themselves. But, they did not. Their intention was not removal of God, rather, to ensure that the government of the United States did not become like the government of England: one church, or "controlling legal authority" spiritually, to which all people are measured and judged.
I agree with your statement, however, government should absolutely not be involved in religion . But not being involved in religion does not mean government cannot recognize a supreme being to which it is accountable and by which it is blessed. George Washington, Abraham Lincoln, and many other of our early presidents and leaders had much to say in their recognition of God and His influence and NEED in our guiding our government, leaders, and people.
Unfortunately, on the other side, others have used their secular public power to advance a religious agenda on the populace, and this too must be stopped. Religion and politics do not mix without disastrous end results (see the Taliban). And it's not always easy to tell sincerety and reason, even for the person doing it.
For example: An Protestant school administrator wants to post the 10 Commandments only because he honestly thinks it's a good code to live by and kids would do well to see it every day. Sounds good to me, not an endorsement of his religion. But both you and I know he'll get very angry when the workers accidentally post the Catholic or Jewish 10 Commandments instead of his. Hmmm, is this administrator really religion-neutral? He meant to be in the beginning.
One also needs to look back to the Founding Fathers and their culture ... But not being involved in religion does not mean government cannot recognize a supreme being to which it is accountable and by which it is blessed.
On your analysis, "One nation under God" and "In God we trust" would most definitely stay a part of our government. However, the Bible and 10 Commandments would be eliminated from our government. Only mention of a guiding higher power, call it "God" if you will, would remain.
This is because most of the Founding Fathers were Deists, a movement popular among the educated of the time. Deists believed in the existence of God on purely rational grounds without any reliance on revealed religion or religious authority. In other words, they rejected the Bible, Qur'an, Torah, etc., but they weren't atheists. This is very well documented, especially in the writings of Jefferson and Madison.