Democracy is deeply flawed as a system of selection for leadership. It effectively follows the socialist ideology - even if it predates Karl Marx by more than 2000 years - in that it gives everyone an equal right to vote regardless of whether the person is educated or ignorant, patriotic or traitor, etc.

Equality is not a socialist doctrine. It is a liberal doctrine and is at the core of the libertarian ideology. Your rights come from your innate equality with everybody else.

It's quite ignorant to suggest equality follows the socialist ideology. Socialism is closer to equity than Equality. Even then socialism at its simplest is workers owning the place they work at. I.e. Miners own the mine, workers own the store etc. In that scenario, there still isn't equity, as the market is still determining the value of goods and thus the take home pay of the workers.