> *12V rail ... 10V pk to peak, which is 3.56V RMS*

The chip is two amplifiers in _bridge_ configuration. You get nearly 4V rms at each output, out-of-phase, so 8V rms across the speaker.

This is roughly 8 Watts in 8 ohms, 16 Watts in 4 ohms. (The chip is rated for 14.4V use, a hot car battery, and specs are taken with significant distortion, so it may say more on the data-sheet.)

> *why do the TG need that much current?*

It is a POWER amplifier, and significant power.

Your 12V 500mA (0.5A) supply only has 12V*0.5A= 6 Watts of DC power. With losses and sine-wave output, a linear audio amplifier can only deliver 3 or 4 Watts of undistorted output.