Australia is not a republic. Canada is not a republic. France is a republic.
A republic is a country that does not have a monarch as a head of state. It has no bearing on how free, or democratic, or representative a country is.
The United States is an (arguably weak) representative democracy, that happens to be a republic.
Realizing that the peasantry needed to be kept away from power was not some enlightened revelation. It's the divine right of kings by another name.
"A number of citizens, whether amounting to a minority or majority of the whole, who are united and actuated by some common impulse of passion, or of interest, adverse to the rights of other citizens, or to the permanent and aggregate interests of the community" is a picture-perfect description of the American political landscape, despite all the hubbub about states rights, direct democracy, and how your representatives know best.
Who exactly nominated and elected Kim Jong-Un? "Eternal President of the Republic" doesn't sound much like a representative of the people to me. It sounds like an emperor who's dishonest about his role.
As in all communist states, Kim Jong-Un was nominated and elected by representatives of the Worker's Party of Korea. Even by Soviet politburo standards, that wasn't a fair election, but formally, the power of the Party Secretary/Grand Poobah/Whatever, is derived from the party.
Unlike most communist states, though, his position is de-facto hereditary. Either way, a strong argument could be made that while the DPRK is a monarchy, the USSR clearly was not one.