My cars’s an SE not EX so has manual A/C not climate control, but there might be a clue here. According to the handbook, one of the conditions that inhibits the auto-stop is if you’re trying to demist the screen, ie you have the ventilation controls set to direct the air flow up onto the screen. But when that’s happening, the right-hand display shows A/C as the reason for it not working: that's wrong, start/stop works perfectly well with the A/C provided the airflow is directed into the cabin rather than up onto the screen. So what setting do you have on that control? And yes, the A/C button lights up when pressed to turn the A/C on.
The start/stop system has a large number of factors that would inhibit it (have a look in the handbook). Some of these are pretty esoteric (eg battery internal temperature), and it’s not easy to work out just which one is affecting the system. Result is that the system is quite erratic, sometimes working, sometimes not, sometimes both in the course of the same journey. Not much help to you, but you’re not alone in finding it frustrating.