Actually, since Washington had no children (and only stepchildren) it might not have worked so well.
Though it might seem odd to think of America as a monarchy, sometimes it seems that way with the way that certain political families pass down power. The Bush family has had two presidents and one governor. The Kennedy's have had one president, one attorney general, one (current) senator, several congressmen, one lieutenant governor, and one governor-in-law. In addition, many family members, despite never being elected anything, are or were beloved. Ex. JFK jr. and Maria Shriver (prior to being first lady of CA), Caroline Kennedy Schlossburg, Eunice Shriver (I'm with Oprah -that woman could have been elected president). There were two Adams presidents and two Harrisons and two Roosevelts. The Clintons are a married ex president and senator. Bob and Elizabeth Dole are a married ex senator and current senator and she used to run the red cross. I know I'm missing others.
One might say that we have had elected monarchies during certain periods in time. I think that the reason we elect so many people who are related is because on some level we like the idea of a royal family -just not enough to get a real one.