Easy answer: B&W 801N or the original Revel Salon, both available within your price range.
The B&W needs to be bi-amped to sound right - it requires an enormous amount of power.
The Salon can go incredibly loud with really high quality bass, and also requires a lot of power, but doesn't require biamplification. If you like 4th order crossover speakers, it is hard to beat, and at used prices, it's an enormous bargain, as it is an excellent speaker.
Large drivers are not necessarily needed to achieve deep, prodigious bass - the quality of the drivers and their implementation is what counts - but yes, larger cones are generally better.
