Not mob rule. That is direct democracy, which often isn't very good. Representative democracy, aka a republic, is quite civilised. Either way yes, that is civilization.
Better to ask, how is it not? It allows a couple layers of sober thought between mob rule and actual rule, while maintaining the collective will of the people.
It is not civilized because it is based on being ruled, whether that be "mob rule" or "actual rule": 100% consent is not ascertained by every single individual; aka, slavery.
Hierarchies are a good thing, and a mark of a civilization. It isn't slavery to have people in charge. Slavery is something very specific. Not just general leadership.
What is slavery? If owing 100% of someone's labor is slavery, is owning 1% of their labor something different? With people in charge, how are those who do not consent, consenting?
No one owns your labour in the current system but yourself. You sell your labour. Someone will purchase your labour if they have a demand for it. What labour you offer matters.