Why is the carrier suppressed by AM-DSB-SC modulation?
It's a simple trigonometric identity:

$$\cos(\omega_mt)\cos(\omega_ct)=\frac12\left[\cos((\omega_m+\omega_c)t)+\cos((\omega_m-\omega_c))\right]\tag{1}$$

Multiplying two sinusoids of different frequencies results in the sum of two sinusoids with the sum and the difference of the two frequencies.

So this type of AM (DSB-SC) really results in a suppressed carrier, i.e., the modulated signal has no component at the carrier frequency. This is true as long as the message signal has no DC component.

What happens is that the spectrum of the message signal is simply shifted to $\pm \omega_c$, so DC gets mapped to the carrier frequency $\omega_c$, and as long as there is no DC component in the message there is also no carrier component in the modulated signal.

EDIT: Your square wave example is perfect. A square wave is a piecewise constant function, so you could say it is just switched DC, or, paraphrasing your question, it's "made up of sections of otherwise unmodified DC". However, if the duration and amplitude of the positive and the negative part of the square wave are equal, then the corresponding signal's DC component is exactly zero. If you accept that, you must also accept that the signal in your example has no component at the carrier frequency.