Conservatism the movement supports Monarchism the system. Liberalism the movement supports Capitalism the system.

It doesn’t. That’s a complete misunderstanding of all four of those terms.

And Communism fights against private property in favor of universal liberation.

I think you have your head twisted a tad.

Liberalism also fought in favor of universal liberation. Not understanding that or how liberalism lead to conversations for socialism is a glaring hole in one’s historical understanding. Either both Liberalism and Communism fight/have fought for universal liberation or, following your logic, Communism fights for a specific system structure as well because “universal liberation” actually doesn’t mean anything you could define. With that in mind it probably would read more like, Communism fights for Authoritarianism.

Communism demands extreme power towards a central point, it relies extensively on what you’d call the state having the first and final say in things. Whether that state is successfully run by “the proletariat” or not is a whole other conversation. But historically stacking all that power in the center begins to circle back towards monarchism maybe without a hereditary line but still you’re handing all power to a single political entity and that’s some heavy Authoritarianism.