My apologies for the delayed response.
The short answer is that the Fourteenth Amendment has been used by the courts to apply rights enumerated in the federal constitution to the states regardless of what the state constitutions might say. Just as elections have consequences, wars have consequences, and those of us who affirm original intent of the Constitution cannot deny that the post-Civil War federal constitution, as amended, gives considerably less latitude to the states than was the case pre-1861.
That is entirely legitimate with regard to slavery, with regard to civil rights and citizenship for blacks, and related matters. The Constitution probably did have to be amended to undo the 1857 Dred Scott decision, and it's much better to make major changes by the formal process of a constitutional amendment than to have shifting majorities on the Supreme Court make such changes based on political considerations rather than the expressed written decisions of Congress and state legislatures.
The longer answer — and this is the crux of the problem — is that the Fourteenth Amendment, whose clear original intent was to bar racial discrimination, has been grossly expanded beyond any conceivable intent of all but the most radical and extreme leftist politicians in the 1860s and 1870s.
While it's true that American religious life included more radical elements in the 1860s and 1870s than it did in the era of the Founding Fathers, and open atheism as well as cultic religion and spiritualist movements with seances were in existence (think Abraham Lincoln's wife, for example) I cannot remotely imagine most Unitarians or Transcendentalists in 1870 wanting to bar religious symbols from public places or remove them from state and city emblems.
Even the Blaine Amendments of the anti-Catholic era of the late 1800s, which were intended to prevent tax dollars from being used for promotion of “sectarian” doctrines in public schools in areas where Roman Catholics were the majority, didn't anticipate that a certain type of lowest-common-denominator Christianity would not be taught in the public schools. The consensus conservative, moderate, and liberal political position in the late 1800s and much of the early 1900s was that religion and morality were essential to a well-ordered society, so while it might not be a good idea to talk too much about the Trinity in a New England public school or about consubstantiation in a public school in heavily Lutheran area of the midwest, religion and morality certainly **WERE** being taught in the public schools so long as it was not “sectarian,” i.e., promoting one denomination over against another.
One needs to look to the role of John Dewey and radical movements of the early 1900s to find any hint of the anti-religious positions which have now become common in jurisprudence. I do not believe that can even remotely be defended with a strict constructionist view of the federal constitution.
Not everything that happened in the late 1800s and early 1900s was necessarily bad — for example, I don't have a problem with the courts extending the long-term allowance of draft exemptions for religiously motivated pacifist Quakers and Mennonites (which date back to the Founding Fathers) to atheists and agnostics who have a philosophical objection to war comparable to religiously based pacifism. I think a very good case can be made that legal tolerance for explicit atheism and irreligion is quite compatible with the Founders’ obvious original intent to allow toleration for Deism, Unitarianism, and other liberal religious systems.
My problem is not that atheism is being tolerated — I have no desire to force people to be church members or profess a faith they do not possess — but rather that the courts are ruling that not only “sectarian” religion but also basic morality must be excised from public actions and discourse.
Nobody can credibly argue that even the most liberal of the Founding Fathers intended that. Not only men like Witherspoon (a Presbyterian minister, president of what is now Princeton, and a right-wing Calvinist by modern standards) but also Unitarians like Adams, and even “freethinkers” like Thomas Jefferson and Benjamin Franklin believed that. The founders were far from unified on their religious beliefs, but they believed morality was essential to maintaining a free republic. They were right, and to pretend that morality does not typically have a religious basis is to ignore virtually all of human history.