If the Catholic Church did not have the authority to decide what was in the Bible and what was not, then the Bible cannot be trusted. There was no "Bible" until around 390 AD. There were many letters, gospels, etc. Only some of them became books of the Bible when the Catholic Church published an official canon. Protestants would have to have a theory that the Catholic Church had authority from God to establish the Bible, but then lost that authority later. The only problem there is that such a theory goes against the Bible itself, which mentions Peter being declared pope (although that term wasn't used at the time), being given the keys to the gates of heaven, and Jesus declared that Hell would not prevail over the Church He established. Protestants must argue that Hell had prevailed over the Church at some point later to declare the supremacy of the Bible over the Church, but the Bible argues against that.
Not to mention that Luther sought to exclude many books from the Bible in the 1500s, over 1000 years after the Bible canon was settled. He wasn't totally successful. Some of the books he wanted removed still remain, but several were removed. He basically said, "I'm the arbiter now." Not a very convincing position.
And, you did ask your question on a Notre Dame forum, after all, so a Catholic bias could be expected.
To your point: I believe you are correct. Catholics were generally not allowed to hold office or even vote, so there were few in the leadership of the new country. Charles Carroll did sign the Declaration of Independence. I think the theory was that religious discrimination by Protestants would be illegal in the new country.