Anyways, in my mind, a real, functioning democracy is where people can directly elect their head of state, which is usually the president

this is such a deficient view of democracy that it doesn't even qualify as democracy in my book. the word means rule by the masses - in what way does electing a head of state, someone who is in no way obligated to respect the wishes of the electorate (cf US presidents and their inability to push legislation/policy supported by supermajorities of the electorate) constitute rule by the masses. merely electing someone passes the buck on from the masses onto singular, corruptible individuals. democracy means the democratic participation of the people in government and such a thing goes so much farther than mere elections for representative. compare Cuban democracy, where elected representatives are required to hold weekly, local office hours so constituents can come share their thoughts about policy and governance, to western "democracy" where most people have no clue who their elected officials even are or what policy decisions their representatives are taking.

dream bigger. you're setting a very narrow horizon on human possibilities in the sociopolitical realm.