It would have been trivially easy to insert language that would admit Protestants and deists (who are farther from Protestants than Catholics, but whatever) into the Constitution or Bill of Rights.

That this was not done indicates that the Founders also did not intend to endorse even a flavor of Protestantism. Heck, they didn't even endorse theism in the text.

For comparison, the Maine constitution was written at the height of the "Second Great Awakening" when religiosity was at a much higher level, and its freedom of religion clause very clearly specifies that it specifically protects the freedom with regard to which way you worship Almighty God.