You can't understand the incoherence of the modern American conservative movement without understanding the contradiction at its core. Burkean Conservatism has as its goal the preservation of the interests of the Good as against the demands of the Many. In eighteenth century Britain this fell right in line with the existing political, economic, and social structures, everyone understood who was High, Middle, and Low and the gradations within those categories. In politics the interest of the High was protected by the House of Lords and the interests of the Middle at least somewhat protected by the House of Commons (because not all the Middle was represented and the High had much influence). But the Low had no representation at all and in the ethos of the time that was a good thing.
In eighteenth century Britain democracy was equated to revolution and specifically the the French Revolution with its radical levelling principles that led to regicide and the breaking of the power of the Established Church. The French Revolution was not just a political, social, and economic convulsion, it seemed in its totality to be a breakdown in cosmic order, a World Turned Upside Down.
The American Revolution started 13 years before the French and so while many of the Founders were sympathetic with the ideas that would drive the later Revolution they were able to implement them in a very different situation, the King not residing right in the midst of them but instead being on the other side of the ocean. Still the Declaration of Independence and the Bill of Rights and the relatively wide franchise were from the perspective of the newly emerging Burkean Conservatism thoroughly radical rejection of the natural order. For Conservatives pure democracy was a threat.
Which puts us where we are. Modern American Conservatism has inherited its foundations from Burkean Conservatism but is constrained by the American political system from proposing them openly. In America you can't openly assert that society is by nature hierarchical and moreover pyramid shaped and that that societal order is best maintained by the control of the 'Good' people in the middle to upper levels of the hierarchy who are by that same nature not only allowed but encouraged to maximize their own 'Deserving' interests against those of the 'Undeserving' 'Many'. That the end result is by definition anti-Democratic is a feature and not a bug.
But in the United States the path to public power comes through obtaining electoral majorities, meaning that the natural representatives of the 'Good' still have to draw the votes of enough of the 'Many' to be first past the post. Which in practice means convincing enough of the 'Many' that they are really among the 'Good' even as they lack the natural economic and social status to actually take their place among the latter. Which has led modern Movement Conservatism to exploit every fissure among the Many, including race, national origin, and religion. For Conservatism 'Good' meant free, white, male and being either wealthy or living in small towns and rural areas and mostly still does. Which makes it an uneasy fit with a country that is increasingly diverse and largely urbanized.
How do you sell Conservative concepts of hierarchy and natural inequality in a system built on the credo that holds that "All Men are Created Equal" is a central tenet of society? Well with great difficulty.