but capitalist system does has centralised authority despite the private ownership of wealth it allows.
Capitalism is not a political system it's an economic model!
Capitalism can exist without a central authority; it has existed since prehistoric times: the guy who had a rock could use this rock and trade his rock for a stick, it's socialism that needs a centralized authority to force the guy with a rock to give everyone in the tribe a rock!
You are right that ideology itself is not, but the people who use the ideology could make it political system. For example, as we are well aware of American political parties, let me give you an example from American parties. Republicans believe in free market, and they are fine with drug companies to charge up to 300 dollars for a simple 3 dollar insulin, whereas democrats think that government should have a chance to intervene and cap that at 30 dollars.
Not a clue on what is the latest situation, it constantly changes, but that is what they believe. In that example, you see that republicans are saying, what democrats are trying to do is socialism, and what republicans aim at is capitalism. Suddenly it becomes a political thing, instead of an economical thing.