Founding era figures Timothy Dwight (President of Yale), Joseph Priestley and Richard Price all thought of themselves as "Christian." Dwight was "orthodox"; Priestley was Socinian; Price was Arian. As it were, all believed Jesus was Messiah and a risen savior. Dwight believed Jesus the Second Person in the Trinity; Priestley, Jesus only man, not divine at all, but on a divine mission; and Price, Jesus, a created but subordinate divine Son, the first created being.
The "orthodox" like Dwight accused unitarians like Priestley and Price of believing, by necessity, in a partially inspired, that is fallible Bible. That's how, the orthodox argue, unitarians derive a non-Triune God from revelation.
With that, here Dr. Dwight discusses Priestley and Price denying the infallibility of the Bible [paragraphs breaks added for clarity]:
... Dr. Priestley says expressly, that he does not consider the books of Scripture as inspired, but as authentic records of the dispensations of God to mankind; with every particular of which we cannot be too well acquainted. The writers of the books of Scripture, he says, were men, and therefore fallible. But all, that we have to do with them, is in the character of historians, and witnesses, of what they heard and saw: like all other historians, they were liable to mistakes.
"Neither I," says he to Dr. Price, "nor, I presume, yourself, believe implicitly every thing, which is advanced by any writer in the Old or New Testament. I believe them," that is, the writers, "to have been men, and therefore fallible." And again; "That the books of Scripture were written by particular divine inspiration is a thing, to which the writers themselves make no pretensions. It is a notion destitute of all proof, and that has done great injury to the evidence of Christianity." The reasonings of the divine writers, he declares, we are fully at liberty to judge of, as we are those of other men. Accordingly, he asserts St. Paul in a particular instance to have reasoned fallaciously; and maintains that Christ was both fallible and peccable.
Other English Socinians unite with Dr. Priestley in these sentiments: while Socinians of other nations proceed so far, as to treat the writers themselves, and their books, with marked contempt. In these several things there is plainly an utter denial, that the Scriptures are a Revelation from God. To all these opinions Dr. Priestley was once directly opposed: for he was once a Trinitarian, and a Calvinist. The inference seems, therefore, to be necessary, that he was led to them all by his denial of the Deity of Christ. A similar transformation appears to have been undergone by many other Socinians; and something very like it by no small number of Arians.
I've long argued that the "Christianity" of Priestley and Price -- what they termed "rational Christianity" -- is closer to what the "key Founders" (Washington, J. Adams, Jefferson, Madison, Franklin, and others) believed than is the "Christianity" of the "orthodox" like Dwight.