"Brushless DC motors" are good because brushed DC motors are constantly switching polarity, which causes arcing of the brushes, which causes wear. The brushes are not there to energize the rotor; the rotor is just magnets after all. The brushes are there to tell the stator to change polarity.

Brushless DC motors don't arc -- because they switch stator polarity with electronics that sense the position of the rotor without rubbing parts. (They can also fine-tune the stator current spikes to make the motor very efficient over a wide speed range, which brushed DC motors cannot do.) The lack of arcing is more important than the fact that they don't have rotating contact points.

Brushed AC motors have rotating contact points (slip rings) but they don't arc (ideally), so the contact points don't degrade as fast as brushed DC motors do. But they do carry a lot of current because their purpose is to energize the rotor. Brushed AC motors are not ideal, but making an AC motor "brushless" is not nearly as big a win as making a DC motor brushless.

Wait. You're saying DC motors require current that's constantly switching polarity? So they're sort of really AC internally?

Yep. All motors require constantly changing current. The distinction between AC and DC motors is whether you feed the motor externally with current that is already alternating sinusoidally, or whether the motor itself turns external DC into some kind of AC.