The quote by Adams is held up as axiomatic, chosen to produce a preordained conclusion, and upon it built a "second reality" wherein all of the Founders share the same religious beliefs, and subscribe to the same sectarian/denominational doctines as the person making the argument.
Balderdash! - Your fear is showing. No one has ever held up a universal, monolithic 'faith of the founders;' Only that they all were indeed believers in "the God of the Bible," which leaves much room for differences.
There is some truth to your complaint, though I would suggest that the Declaration's "we hold these truths to be self-evident" is the part more properly identified as being "held up as axiomatic." On what other basis, after all, can the rights life, liberty and pursuit of happiness properly be labled as "unalienable?" They are axiomatic in the sense that they are "endowed by our Creator." I think the left doesn't actually believe this to be true -- at least, not in the way the Founders did, and also not like at least some conservatives do.
In that case, of course, the objective basis for the correctness of the Declaration would depend on a "first reality" that actually includes a Creator.
I like the Adams quote not for any "axiomatic" reasons, but rather for its practical (if unspecific) statement of the necessary conditions for limited government. Absent the self-policing nature of a "moral and religious people," it is difficult to imagine a system of "limited government" as not descending into anarchy; or, in order to prevent anarchy, the government would tend toward tyranny.
The question for you is, is there anything that you could offer as a "basis for conservatism" that is not in some sense "axiomatic" in a "second reality" sense?
John Adams' insight is neither sectarian nor doctrinal. What it represents is a deep insight into human nature shared by all the Framers regardless of religious confession. It has deep roots in history, philosophy, and culture, and was a major concern for Plato, who saw that no political order could be any better than the general moral "tone" of the people who compose it.
The Constitution was designed for a free people who are morally responsible for their actions. When we speak of a system of self-government, which is what we in America supposedly have, we have to recognize that "self-government" begins in the good order of the individual citizen: Personal morality is the foundation of the system. If the people are "disordered," then so will be the society. And the Constitution itself eventually will come under attack.
I believe that is the point that John Adams was asserting.