I disconnected my battery for 30 minutes at the weekend, the stop start was working but intermittently, doesn't bother me as I switch it off anyway, now it's far more consistent if I forget to press the button to cancel it.
The car will disable the stop start during regeneration and have a slightly lumpy idle, I have a diagnostic connected as I'm currently monitoring regen intervals after fitting a genuine dpf sensor.
Without a diagnostic you might struggle to notice when it's regenerating.
As said a low battery voltage or battery on its way out may also disable the stop start, there's also a couple of other things it monitors on the car but can't remember what they are.
I've had problems with the car regenerating too often but not showing any codes, it's a long story but I'm hoping it's fixed now with a genuine hyundai sensor.