It's debatable whether America knew the "true" (perhaps heterodox) Locke. And Priestley and Price weren't popular among the masses, but rather influenced various elites in important positions of power (they greatly influenced J. Adams, Jefferson, and Franklin).
Thomas Gordon & John Trenchard's works were more popularly read in America. They posited something which most (if not every) of the above mentioned figures endorsed -- an idea of "primitive Christianity," that is Christianity before it got corrupted by clerics and creeds.
Trenchard and Gordon's "test" for what is "Christianity" seems close to John Locke's (who certainly influenced them). There are two elements, one belief in the Bible as God's divinely inspired word and two Jesus as Messiah and son of God. Nothing contained in a creed is necessary. In fact, ecclesiastical authority and official creeds tend to be hated in this kind of Christianity. Roman Catholicism is enemy number one. High church Anglicanism is enemy number two. But again, the notion of creedalism itself is rejected.
Anti-creedalism is something I've discovered quite common among America's Founders. Not just the "key Founders" like the first four Presidents and Ben Franklin, but also figures like Benjamin Rush and John Jay. We might term this kind of Christianity, "freethinking Christianity" that contrasts with the more conservative "creedal Christianity." Where Protestantism meets Enlightenment, the freethinkers wanted to read the Bible for themselves and determine what it meant. The Bible clearly refers to Jesus as Son of the God and Savior of man. But does it clearly teach the Trinity?
The freethinking Christians say no. Indeed, I realize a lot of Sola Scriptura Protestants claim they get the Trinity from the Bible alone. However, if you research church history and experience, you'll see, almost without exception these churches likewise embrace orthodox creeds like the Athanasian, Nicene, Apostles, because they realize freethinkers reading the Bible for themselves will NOT necessarily conclude that it clearly teaches the Trinity. Therefore those creeds are necessary just to make clear THIS is how we interpret the Bible.
Trenchard and Gordon are exhibit A.
As they wrote Wednesday, April 6. 1720:
But whilst [the clergy] were thus carrying on their Project for Dominion, they found it necessary to throw out a Barrel to the Whale, and keep the People’s Minds busied, and their Passions afloat, with Metaphysical Subtilties and Distinctions, of no Use to true Religion and Morality, though very conducive to their own ambitious tyrannical Designs.
I would gladly know, from these Reverend Venders of Trifles, Whether it would have been worth the Thousandth Part of the Combustion which has been made, or the Blood which has been spilt, only to have settled a few Speculations, if they could have been settled? Pray where is the essential Difference between Transubstantiation, Consubstantiation, and the RealPresence? What the Consequence, whether a Child be baptized by one sort of Priests, or by another? Or of what Use to Mankind are the abstruse Questions about Predestination, Free-Will, or Free-Grace? What is the Difference, as to the Duties or Ordinances of Christianity, if they be administred under the Direction of a single Person, a Bench of Bishops, or a Lower House of Convocation, or none of them all, so they be piously administred? Or whether the chimerical Line of Succession be broken, or ever had a Being?
Since ’tis agreed amongst all our present Sects of Christians, that the Saviour of the World is the Son of God, descended from Heaven to teach Virtue and Goodness to Men, and to die for our Redemption; how are we concerned in the Scholastic Notions of the Trinity? Will the Scripture be more regarded, or the Precepts of it be better observed, if the Three Persons are believed to be Three Divine distinct Spirits and Minds, who are so many real subsisting Persons? Whether the Son and Holy-Ghost are Omnipotent of themselves, or are subordinate, and dependent on the Father? Or, if they are independent, whether their Union consist in a mutual Consciousness of one another’s Thoughts and Designs, or in any thing else? Whether they are Three Attributes of God, viz. Goodness, Wisdom and Power? Or Three internal Acts, viz. Creation, Redemption and Sanctification? Or Two internal Acts of the One subsisting Person of the Father; that is to say, the Father understanding and willing himself and his own Perfections? Or Three internal Relations, namely, the Divine Substance and Godhead confidered as Unbegotten, Begotten, and Proceeding? Or Three Names of God ascribed to him in Holy Scripture, as he is Father of all Things, as he did inhabit in an extraordinary Manner in the Man Jesus Christ, and as he effected every thing by his Spirit, or his Energy and Power? Or lastly, Whether the Three Persons are only Three Beings, but what sort of Beings we neither know, nor ought to pretend to know? which I take to be the Trinity of the Mob, as well as of some other wiser Heads.
As far as I can remember, these are the important Questions which have set Mankind together by the Ears, for so many Ages; and it seems are yet thought of Consequence enough to create new Feuds, and mortal Dudgeon, amongst all our Sects of Ecclesiastics. But why must we of the Laity quarrel about them too? What have Beaux and Belles, old Women, Coblers, and Milk-Maids, to do with Homo-ousios, Consubstantiality, Personality, HypostaticalUnion, Infinite Satisfaction, &c.? none of which hard Words, or any like them, are to be found in Scripture; and therefore, I think, we may even return them to Rome, that being the Place from whence they came, and be contented to be good Christians without them.
Something else we see here is the notion that orthodox Trinitarianism itself, or at minimum many of doctrines which exist in orthodox creeds and confessions, becomes associated with Roman Catholic fabrication.