Conservative and liberal capitalists suffer under two differing delusions: conservatives that capitalism is compatible with "tradition", the liberals that capitalism is compatible with democracy. Neither side suffers the opposing delusion.
Because right capitalists don't really believe in democracy they will throw it out at the slightest sign of anti-capitalist power. The only real difference between fascism and conservatism is the presence of an anti-capitalist threat.
(This is why Trump is not a fascist. No anti-capitalist threat! Allen Dulles and J. Edgar Hoover were fascist than Trump was. They had actual lefty threats to fight!)
Right capitalists also embrace things like family, tradition, culture, and nationalism, which they view as necessary to preserve society. And they think these things can be maintained under capitalism, that they are in some way necessary to preserving capitalism!
Of course capitalism doesn't actually care about these things like tradition and culture and family and nation. Under capitalism everything solid melts into air, all is disrupted in the name of innovation, productivity and profit.
Liberal capitalists are more comfortable with this. They are ok with the times changing, religions falling away, genders breaking down, races intermixing, the borders of nations loosening. This is progress!
But liberals thinks this progress can equalize society, give everyone rights and eliminate hierarchy. Except that capitalism, as a class-based economic system, is inherently heirarchal. As it dissolves social hierarchies of race, religion, and gender, class will remain.
And where class remains, where hierarchy remains, democracy is impossible. Democracy is ultimately not compatible with class hierarchy, or with the exploitation that hierarchy serves, for if given a real option, free of hegemonic ideology, no worker would choose exploitation.
For a bit, some left capitalists resolved this problem, when they briefly became aware of it, by pushing for "social democracy": keep capitalism but give workers just enough to get by comfortably. This might have worked if not for the continued existence of right capitalists.
Eventually the right capitalists maneuvered so as to wipe out social democrats (IE New Dealers) out, along with anti-capitalists, and we entered the neoliberal era, a period of seemingly ascendant right capitalism, with a neoliberal left capitalism replacing the New Deal types.
But that era feels like it's at a close, because while the right neoliberals seemed in control, there was capitalism, unbridled, eating very quickly away at all those traditional social bonds and hierarchies the right capitalists loved. Hoist by their own petard!
Look at the capitalist class. The real billionaires now in control. They are all liberals! But do they believe in democracy? Well...
It seems like we might be living through a synthesis of the left and right capitalist dialectic. We might be seeing a new capitalism emerge, a kind if true neoliberalism, unbound both by any delusions of tradition, but of democracy as well.
With the rejection of Trump and his reactionary base, Capital seems to have, in the Biden administration, found a kind of liberal, multicultural, diverse, tolerant, and authoritarian capitalism.
The irony of course is that this neoliberal multicultural authoritarianism is stuck sharing a political party with a small nascent anti-capitalist movement, as the second political party is still held by the fading right. Obviously this situation cannot endure indefinitely.