Some ramblings about the denomination of my birth, which didn't see itself as a denomination. As I think Msgr. Burnham recently mentioned, in England it really thought of itself, Catholic-like, as "the church" (as I intentionally, habitually call the Catholic Church), the country's other major Christian groups being "Romans" or "nonconformists/dissenters." Catholics were considered "Catholic too" but in grave error; other Protestants seen much as we see them ("No bishop, no ministry!").
Michael Davies for example has taught me that Luther and the original Anglicans really weren't big on continuity with the medieval church; any resemblance was either accidental or a cover to promote the new faith, which of course they saw as a long-belated recovery of the original from Christ. Luther kept externals and Melanchthon tried most of his adult life to reconcile Lutheranism to the church so Lutherans ended up our close cousins.
Anglicanism is Reformed, a label that doesn't necessarily mean Calvinist. A few generations after the King forced England's separation from the church, the high churchmen claimed the new religion was the old religion minus "accretions," complete with real bishops so the church fathers were writing about the high churchmen's faith, neither "Roman" nor nonconformist. (The high churchmen according to Jonathan Mitchican: the true church, thanks to being both Catholic and Reformed, "Catholic" meaning creeds, bishops, and the idea of a liturgy, not a "branch" of the church and certainly not wannabe Catholics.)
An analogy to classic cars comes to mind: they're claiming they've customized and streamlined the car (no more drag from those "Roman accretions") but it's still a real '58 Chevy Impala (to use a favorite example: the bigger, cooler, double twin-headlight cars, not the cliché Tri-5 Chevys, which look like taxis to me), for example, original "guts," engine, drivetrain, and all.
It's more like an engine swap and more, exactly the opposite, plus the cosmetic changes, which in this case reflect the new faith. If the body shell's a '58 Chevy but a modern hemi and transmission are powering the car, is she really still a '58 Chevy? Not enough to be registered in Pennsylvania as an antique.
In any event, while there was enough remaining orthodoxy to give people a good start in Christian formation (happy historical accident: why Catholicism in English speaks to me in its idiom and why I'm not Novus Ordo), it was Erastian, solely to give the King what he wanted, so no wonder the English elite (and their cousins, America's founding fathers), having pillaged the church locally thanks to the change in religion, lost their belief when the "Enlightenment" hit.