I really don't like the pushbutton start, a real key gives you a tactile feel. I'm guessing the gasoline engine restarts to keep the AC / heat going.

I place the push start up there with the 80's Buick Riviera touch screen that controlled the radio / hvac, Chryslers talking cars , Fords horn on the turn signal stalk , daytime running lights introduced be GM that use the headlights ( and wash out the turn signals ) and the clear turn signals that are invisible in strong light ( yellow lenses will glow where the clear ones don't. ) .

One fix would be to install a light at the top of dash / grille area showing that the car is still "on". Also check the parts dept to see if a real key conversion exists.