Hello99,
I don’t think it’s much of an extrapolation or dochek. The R’ma clearly assers the food b’dieved if it was heated after having cooled, although it’s pretty clear to me that you’ll be stuck without (hot) food.
I think these halachos (chazara, bishul) are more chamur than the case of the Itur. Besides the sevara that we must be more machmir for food, your being neheneh directly from his maaseh (the food itself was cooked by him) which might be worse than using the lamp he turned on.
What is your alternative? That the R’ma would allow full-fledged bishul al y’dei aku”m when you have no other cholent, but asser food even b’dieved when it was reheated after being nitztanein?