I wasn't on board with the idea of a draft either...but part of what doing a draft does is forces people from a large cross-section into the military. Even if the rich and influential people manage to keep their kids out, you now have voters across a large spectrum who are in the military or have kids in the military.
Because we've shown the American people we'll do any job, no matter the cost or lack of resources, and because so few Americans have contact with the military, we are now the answer to every problem. Libya? Send the military. Afghanistan? Send the military. Iran? Military will take care of it. The American people are far too willing to commit to military action because, on the whole, it costs them nothing. Their kids aren't in. It's all financed with deficit spending, so their taxes aren't going up. There are no food or steel or rubber shortages. It makes the American people apathetic when our political leadership wants to use the military for every minor problem around the world.
Part of what convinced our government to end Vietnam was the fact that draftees came from a wide-range of backgrounds, creating a wide-spread opposition to sending our military to die in a rice field thousands of miles away for nebulous objectives. Now, most Americans couldn't find Iraq and Afghanistan on a map, and most think we're done deploying.