I don't think Plato's Republic would match most modern definitions of "democracy"

No, Plato sketched out the way that his Republic would gradually deteriorate - one of the inevitable stages was "democracy". He also suggested that once the people began to rule, they would not only enjoy their freedom, they would begin to value freedom for its own sake - and then they would start to do ridiculous things, like free all the slaves, allow women to rule, and even show concern for the rights of animals.