Here is an easier example. Take three metal rods being warmed by constant independent sources. You have a metal rod warmed to 80 C in a room at 30 C. You take two other metal rods warmed to 50 C and connect them to the 80 C metal rod. That will not increase the temperature of the 80 C metal rod. It will decrease the temperature of the 80 C rod and increase the Temperature of the 50 C rods. Exactly as the flame and ice cube example. Or exactly as the Sun and Earth example. Energy flows from warm to cold. Always. From high excitation to lower excitation. Not the other way. Used to think it could, but have recently been convinced it cannot.
That is true. The emission and absorption spectra apply to particular temperatures. If the food is warmed from 300 to 301 the spectrum will be a little higher and stretched. As you say, some of the heat will be absorbed by chemical changes or state changes in the food.
You cannot make something being warmed by absorbing visibile light,
Things warm in sunlight, both from the infrared and the visible. Plants and material inside a greenhouse warm from sunlight, and that is partly from the visible light and partly from the glass which absorbs the solar infrared.
You cannot make your sunburn worse with red light.
Sunburn is caused by radiation forming a new bond within a molecule. It is a molecule changing to a different kind of molecule, not a quantized energy change. The more obvious answer is that you cannot reverse sunburn by giving off UV.
You take two other metal rods warmed to 50 C and connect them to the 80 C metal rod. That will not increase the temperature of the 80 C metal rod. It will decrease the temperature of the 80 C rod and increase the Temperature of the 50 C rods
That is conduction, not radiation.