How is the signal Multiplied with the Carrier in AM
But this seems like an awfully complex, expensive and bulky solution
  for something that needs to be done often


Would you say the same about the Exclusive OR gate? It's a device that performs multiplication on digital signals and is the founding design for the Gilbert cell: -



Picture source.

Would you say that using a Gilbert cell for an EXOR gate is "an awfully complex, expensive and bulky solution for something that needs to be done often"?

Of course if you wanted a really simple AM modulator you could use a couple of diodes and a filter and get something that was quite linear enough for AM. You could extend this to a four quadrant mixer using more diodes or you could go the whole hog and use a gilbert cell and filter.