The Establishment Clause is something of an outlier in constitutional jurisprudence.
First, the Establishment Clause, as ratified in 1791, was a federalism provision. It prevented Congress from interfering with state establishments of religion. That provision suggests that states can have established religions, and that Congress cannot disestablish any of those religions. As all know, there were several established churches at the time of the founding. And if tradition means anything, then practices from 1791 are directly relevant to this question.
Second, as Justice Thomas has observed, such a federalism provision resists incorporation. I've not seen any compelling evidence from debates during the 39th Congress suggesting that the Fourteenth Amendment would place the states under the strictures of the Establishment Clause. (And if the Section 3 debates has taught us anything, the only relevant speeches that matter to understand the Fourteenth Amendments are remarks from congressional republicans in a narrow two-year window.) To be sure, there were comments about the freedom of conscience, perhaps as a privilege or immunity of citizenship, but not about the Establishment Clause in particular–and certainly not about how the Warren Court understood the Establishment Clause. And by the 1860s, all of the Established Churches had been disestablished. But as a practical matter, it was never clear to me how the Establishment Clause could even be incorporated as a privilege or immunity of citizenship, or even as a liberty protected by the Due Process Clause.