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I think you can guess what side of the ‘Pond’ I’m on. The date is deliberate as there was considerable support for the Declaration of Independence as it was felt the ‘state’. i.e. the Crown (not the King himself) - the Crown and monarchy are somewhat different institutions here and nebulous - was exercising too much control over peoples’ lives. When it comes to the notion of a separate ‘state’ that is secular, I understand the Constitution and Bill of Rights to mean that there is to be no ‘establihed’ state religion as there still is in England - The Church of England.
The extent to which the church is established here is witnessed in the remaining and slightly embarrassing fact that 24 bishops of the church (Lords Spiritual) sit in the House of Lords and can comment on, suggest amendments to, and even actively vote down proposed legislation. It was the prevention of any such system or influence upon the determining of good governance by and for the people that Jefferson et al sought to ‘separate church and state’, although without necessarily in so few words. Everyone at the time would have been in no doubt of the aims of the Constitution and Bill of Rights in the same manner that everyone is clear what the aims of the 2nd Amendment seek (even if we argue over the practical implications).
I also find it interesting that the interpretation of the Constitution does now largely reside with the SC and not both Houses of Congress. This is the route England took after Magna Carter, when precedent and interpretation came to represent the ‘unwritten’ constitution here. We are still discussing nearly a 1000 years later whether we should have a single framing constiutional document. Many of us also want the Church of England disestablished (why should the PM appoint bishops using the Royal Perogative?). There is, arguably, a place for religion in society, but not in the exercise of government.