Do commercial AM radio stations broadcast a carrier?
AM broadcast stations do exactly that: broadcast AM.  Not single sideband, double sideband with suppressed carrier, or anything else.  That means a carrier in the middle and a sideband on each side.

The fact that the carrier is a significant component of what is transmitted should also be obvious from a few seconds thought.  By definition, AM changes the amplitude of a carrier to encode the baseband signal.  At full or 100% modulation, the carrier amplitude varies between 0 and some maximum, with half that value being the average.  Obviously there is significant carrier component in the result.