The difference between the democracy and the republic is that the later is bound by a constitution.
Republic requires a Constitution that sets limits to the power. Republic does not require senators - our Constitution requires Senators.
The US Constitution sets a Congress - in the congress the majority rules (democracy). The laws produced by the Congress have to obey the Constitution (Republic).
Republic does not ask for politicians, but for document that sets limits and rules for the powers.
It is OK to say that my Virtual Congress is unconstitutional today. But there is no problem for it to be part of a future Republic.
China is also a Republic - just with a crappy Constitution.
I agree that direct voting is against our current Constitution.