Certainly sounds like there’s a fault then.
FWIW Stop/Start worked perfectly on my pre-facelift car, only inactivating itself at low temps, or at other times you’d expect. On my facelift car it has a mind of its own, just like yours, with apparently no reason why it decides to either work or not work. It’s had the dealer look at it multiple times, and over the last couple of years they’ve replaced various sensors and both batteries under warranty, but it’s still very erratic and now the car’s out of warranty so that’s it.
One thing that I did have for a period was that the ECO indicator would be green in the instrument cluster until I almost came to a stop at which point it would turn yellow. That was cured by replacing the gearbox auxiliary oil pump.
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