> I don't think this would work in a country. In my opinion, democracy only makes sense when the state of the entire set of decisions can be held by every voter's mind. [...] Republics work well because they break up the required duties among the people, so the amount of work required to vote is decreased.

Delegation in liquid democracy solves this problem in much the same way that representation in representative democracy does. "Republic" is really neither here nor there: a Republic can, like the US, be a representative democracy or it can, be an autocratic dictatorship or just about anything else, so long as its not a monarchy. (Conversely, a constitutional monarchy can be a representative democracy in substantive form without being a democracy, e.g., as in the UK.)