The use of representatives isn't just because direct democracy is unfeasible, it also provides a dampening effect, and hopefully provides a foothold for experts to weigh in alongside public opinion.
But it's even worse than that. Because it is seemingly unfeasible for the legislative body to write the specifics into laws, they create unelected bureaucracies to implement the specifics of broad reaching policy goals.
My only real problem with the representative system of democracy generally, is the way we have to proxy our voting authority through our political leaders. It just seems archaic and ludicrous for a region of people (sometimes many hundreds of thousands of people) to have to boil down their political ideologies to a single "best fit" candidate. Even ignoring the potential for corruption, personal motivations, hidden biases etc. you've simplified thousands of separate beliefs on an equal number of issues, social, economic, all, into ONE PERSONS' beliefs and worst of all...that person then has a legitimate claim to believe that they're representing the people.
I suppose it's the difference between a politician believing that they're a "representation" of the people of the electorate, OR a "representative" of those voters. I think most politicians consider themselves the former, and I think that's a mistake.