I thing what Democrats are conserving in simply the status quo of society.

They want politics to be about electing a party with votes, districts

They want banks to organize money.

They want corporations to employ people.

They want schools to be about preparing kids for work and freeing adults to do labor.

They want the global appearance of their finances look good. (GDP)

They want to be against war and violence except if it is in their own benefit.

They want to not care about existential issues like climate change.

I am not saying here that all of the above are overdue change but more and more appears fundamentally broken and there is no ambition for real change.

They may have meant conservative in an older sense of wanting to change things, but slowly. American "conservatives" are more properly called reactionaries because they neither want to keep things the way they are, nor incremental progress (both of which fall under conservativism), but rather want to actively roll back and take away rights that have already been won. Liberals are conservatives because they want gradual, incremental change while preserving the social order. Conservativism is still right-wing though so that part of their statement is incorrect.

Did they not confirm that because they’d immediately show themselves to be wrong?

Do democrats not view a hierarchical social order as inevitable, natural, normal, or desirable? Capitalism is a hierarchical social orders and liberals definitionally support capitalism.

Why are you so obsessed with American politics? What is wrong with you?

Capitalism. And if someone is an anarchist, like me, the state. Those are the two biggest things. And capitalism is really the big kicker.