It was a split between traditionalists and rationalists.
America did not have traditionalists. The split that developed was between two schools of rationalists - doctrinaire liberals and practical liberals.
In Europe the traditionalists of Britain were eliminated by the Civil War and the Glorious Revolution. That was 1688. In France it was the Revolution in 1789. In Spain and Italy it was the Napoleonic Wars. In Germany it was the revolutions of 1848.
The Reformation, the source of radical individualism, had originally set all Leftism in motion.
In any case, the civic institutions that conservatives wanted to preserve had all been destroyed or repurposed by 1850.
That created a social void, one which socialism, communism, syndicalism, anarchism and fascism tried to fill - by manufacturing ideologies that would create the kind of social cohesion that had been fragmenting for centuries.
Fascism differed from Communism only in this: it wanted to use any remaining stones of the social edifice - religious faith, nationalist fervor, noble heredity - as tools to build a society that would allow its ideologues to gather complete power to itself without any accountability.
Communism may have technically abolished private property - but all that meant in practice was that enemies of the regime were expropriated and friends of the regime were assigned the spoils. Communism may have been officially atheist, but Stalin's relationship with the Orthodox Church, Mao's relationship with the churches, etc. was toleration of believers who would accept total subservience to the state.
Fascism was, despite revisionist claims, just as anticlerical as Communism. The Concordats between the Church and the Nazi and Fascist regimes were agreements by the Church to accept official abuse, humiliation, and outright theft in exchange for preserving the liberty of the Church to preach the Word. That promise was not kept.
Fascism is not conservative - not because it rejects atomistic individualism - but because it is the worship of earthly power rather than worship of the living God.
Close, but IMO not quite accurate.
There was an ancient tradition in England, much older than but first written down in Magna Carta, of the right of "free men" to in general live their lives without interference by the King. "Free men" originally meant, more or less, the nobility, but the definition gradually expanded over the centuries. In this tradition the King is not a semi-divine absolute ruler, he is merely the first among equals of those witharistocratic freedom. He has morally and legally enforceable obligations to his peers, who have the right to force him to honor him.
This same tradition is found in all other European countries, with nobility, estates or free towns, etc. struggling for freedom against the King, Duke or whatever.
With the advent of gunpowder weaponry, particularly artillery, such ad hoc coalitions of "free men" lost their ability to content militarily against the King. Artillery just cost too much for private individuals.
Those nations that refused to abandon their principles of independence of "free men," such as Poland, got squashed by neighbors who did. This led to the rise of absolute monarchy throughout Europe except in Britain, which for obvious reasons was not faced with a direct military threat. All it needed to defend itself was a Navy, which could not be used directly to control the citizenry. The tradition of aristocratic freedom thus survived (nearly) uniquely in England.
Absolutism eventually called up opposition against itself in Europe, but since the tradition of aristocratic freedom had been destroyed, the opposition was intellectual in nature, and based on opposition to King and Church, and a mystical belief in the General Will of The People. Little or no interest in individuals within that People.
Meanwhile, in the Civil Wars and then in the Glorious Revolution, the tradition of aristocratic freedom (as expanded) triumphed in England and eventually the whole UK.
When the English colonies in America were colonized, the social pyramid was amputated on both ends. Few peasants or nobles emigrated. The colonists thus thought of themselves as "free men," and assigned to themselves all the rights originally restricted to aristocrats.
Our Revolution was very specifically and explicitly initiated to protect that tradition of (originally) aristocratic freedom against what American saw (inaccurately) as conscious attempts by the British government to take them away.
Anywho, my somewhat long-winded point is that the American Revolution was indeed profoundly traditionalist and conservative. Except the tradition it tried to conserve was that of Magna Carta and aristocratic freedom. That tradition, expanded to cover all adults, is still what American conservatives are trying to conserve.
What you see as the “traditional” society is apparently that of the early modern period, say 1500 to 1700. But that was itself a society in profound turmoil, since economic and military changes had destroyed the more or less stable traditional society of the Middle Ages.
Economic and (especially) military conditions ensured the destruction of that earlier society. The nobles that fought to protect the (local) peasantry and churchmen utterly lost their role when gunpowder weaponry and especially artillery made them obsolete. With the nobility having lost their function, it became an extremely obvious question why society should continue to pay them forever the honors and income their ancestors had legitimately earned.
The rise of cities and of people who were not peasants, churchmen or warriors was also profoundly disturbing. They grew in numbers, wealth and influence throughout this period. Many merchants and businessmen became much wealthier than many nobles.
My point is that changes in society undermined the traditional order. Something was going to replace it. In America we developed a system based on the (originally aristocratic) right of liberty for individuals.
That POV never got on really big in Europe, resulting in their many experiments with Jacobinism, communism, fascism, etc.
This is correct. However, I would argue that some of these ideologies were clearly left-wing insofar as they rejected all aspects of traditional society (private property, title, religion, and in many cases national identity) while others were right-wing in that they rallied in the name of nation, rank, and tradition.