Because it isn't actually a good idea. The Bush administration wasn't dominated by the extremist deficit hawk end of the Republican party. In fact, they had the opposite problem, they ignored the deficit far too much. During good economic times, as we had in early 2001 and 2005-2007, for example, paying down the debt is among the biggest goals that the administration should have. During bad economic times, job creation should be number 1 and that, unfortunately, requires deficit spending. Because deficit spending is often necessary, the majority of law-makers, who understand these things, don't want a balanced budget amendment. The ones who advocate for it are demagogues who know it will never pass and are using it as political fodder for a public who doesn't know enough about macro-economics to see the potential problems.
If you want the easiest argument against the balanced budget amendment, let's look at WWII. If we had a balanced budget amendment, it would have been entirely impossible, absolutely and utterly undoable, for the US to enter the war and fight Hitler. Those kinds of extreme needs can and do arise and the US needs to have the ability to respond to them, regardless of our fiscal situation at the time.