If you think the way that A/C button and indicator mean "the A/C compressor is readily available to use", then you'll be able to follow the logic behind it. This is not the Mazda way; as the matter of fact, every "Automatic Climate Control" system with a A/C button will act this way. Although the A/C indicator is on, that doesn't mean the compressor is turning or the cooling is in work. Compressor is controlled by several sensors, not just by your A/C button. For example, if ambient temperature is 40F and you set "Automatic Climate Control" temperature to 68F. If you push the A/C button and its indicator is on, the compressor is not necessary turning depending on several sensors' feedbacks; i.e. if you set the Mode to defrost then the compressor will be turning. But the heated air is guaranteed coming in. Since the A/C button on "Automatic Climate Control" is confusing to its purpose, VW changed the name of the button to "ECON" on its "Automatic Climate Control" system.
Back to old days, you push the A/C button, the compressor is almost guaranteed on. But with more complex Automatic Climate Control nowadays, this is not true anymore. On the other hand, if you want to make sure the compressor is off no matter what, then push the A/C button to make the A/C indicator disappear. That's why VW calls the same button "ECON" which may make more sense.