Honestly that exact thing happened to me last summer. Tried to sear some chicken in a place with a terrible vent hood. The whole floor got that lovely burnt dinner smell and the alarm would not stop. Felt like I needed to apologize to an entire city block. Those rental stoves are a special kind of unpredictable.
I mean, was it actually the whole building though? Those alarms are super sensitive in rentals. I've set one off just making toast before, and it was only my unit. Idk, maybe it's just me but getting a little smoke doesn't feel like a huge crisis. You just wave a towel at it and move on.