‘Conservatism’ is not a monolithic philosophy. There are social conservatives, economic conservatives, Hayekian conservatives, etc. Also, not all Republicans are conservative, and not all conservatives are Republican.
Hayekian conservatives (although Hayek himself was not a conservative) believe primarily that central planning is a bad idea. They believe that local knowledge is important, that emergent, evolved structures are generally better than command structures, that voluntary cooperation is better than mandates, and that central authorities do not have the knowledge required to make good decisions for distant people. Attempts to plan and control a society ultimately fail due to the unintended consequences of trying to control a complex adaptive system.
The conservative impulse is to say, “Don’t mess with what has evolved, as you probably don’t understand why it evolved this way, and your replacement for it is likely to be inferior or have unintended consequences that will bite us in the ass.” Evolutionary biologists are ‘conservative’ that way. They wouldn’t dream of forcing changes to the ecosystem to make it ‘better’. Hayekians extend that to human society as a general principle - with exceptions where absolutely necessary.
There is absolutely nothing bigoted about this. In fact, many conservatives believe that minorities have been greatly harmed by left-wing policies that had good intentions but terrible results in practice.
Another conservative principle is that power should be held closest to those affected by it. That means first the individual should have freedom and agency to act. If that’s not sufficient, local community is better than regional government, and regional government is better than federal government, which is better than world government.
Then there is the fundamental break between Locke and Rousseau, with Locke basically stating that people are born free, and have a right to live for themselves so long as they don’t hurt others. Rousseau, on the other hand, was a communitarian who believed that people are born into communities with responsibilities to the community, and the community has responsibilities to them. Think of it as, “Rugged Individualism” vs “It Takes a Village”.
Neither of those are at all bigoted positions, but I’d argue that individual rights are the best way to protect minorities, and the leftist project of gathering people into identity groups and treating people as group members rather than individuals is far more dangerous to minorities in the long run, as the people in the minority groups are only safe at the whim of the majority when individual rights, freedom of speech and association and other classically liberal values are shunted aside in favor of identity politics.