I ordered an intumescent paint additive which adds fire retardance to any latex paint, mixed it into a can of white primer, and gave it a quick test. It works!
The stormboard on the left was painted with plain latex primer. The stormboard on the right was painted with the same primer, with an additive called Flame Stop III mixed in. They were both exposed to a candle flame. The untreated board is shown after 90 seconds, when it ignited and sustained flame. The treated board is shown after 180 seconds, and still hasn't ignited.
Intumescent coatings work by forming a foam at high temperature. In a thick industrial coating, the foam would be inches thick and really fireproof the structure. In a thin paint the defense is minimal but significant.
We'll use this primer in the kitchen, behind all the appliances and cabinets.
Other fire-safety measures include backing the refrigerator, dishwasher, and stove cavities with rock wool, and wrapping bathroom fans with UL94-V0 nitrile foam. These materials are all good sound absorbers, as well as fire retardant.
Then there's all the building-code measures, like sealing floor penetrations with fire retardant caulk and foam, hardwired smoke detectors, egress codes, etc.