The United States doesn't have a democracy for many reasons. One reason is the denial of the right of many people to influence the direction of the country based upon where they live.
The reality is that some voters are considered more important than others, a practice which cannot be tolerated if equality is upheld.
A fair government would give all people the meaningful right to vote, but the nation doesn't do so, because if you live in certain "red" or "blue" areas, your vote is meaningless.
Representative democracy might be convincing for representative issues, like how to allocate funds to cities, and it could ensure that people from different areas are represented fairly in that regard. But what it doesn't do well is allow people equal influence on policy directions.
We should not use a representative system for national policy.