Adding a constant to a bandpass-signal in transmission
You're right, from a pure information point of view, the constant just wastes transmit power and doesn't add anything to the signal that you'd care about – the receiver will filter it out.

However, you've got to realize that all AM audio transmission schemes are positively ancient. So, receiver's abilities to demodulate a signal were very limited.

Simple (analog) AM receivers are nothing but a frequency selective circuit (e.g. a tuned antenna) and a rectifier, and a low-pass filter – done. If you're off a bit when tuning the circuit, it doesn't really matter.

Now, in your stereo AM (never seen that, wouldn't do that, seems strange and wasteful in spectrum) transmitter,  it's very important to tune to the right frequency for the first and second channel to be correctly received. 

Now, that requires something a simple receiver could "lock" on. And that's exactly what the constant is: It adds a constant peak in the spectrum. If you have something like a PLL, you can lock on to that (make the PLL's bandwidth small, so that it's not affected by the AM).

Modern receivers could be much smarter – it's not very hard to guess the "center" of a double-sideband AM spectrum without a pilot tone.