bubble wrap doesn't stay inflated forever. Think about what happens to a balloon over time. Bubble wrap doesn't lose air as fast as a balloon but it will eventually
I don't buy it.
Balloons filled with helium deflate rapidly because helium fits right through the latex.
Balloons filled with air deflate because the stretched balloon creates pressure on the inside. Once they have deflated to the point that the latex is no longer stretched, they will not deflate any more.
Bubble wrap is not pressurized. Yes, the plastic acts as a membrane that lets air out, but with equal pressure on each side of the membrane, air will enter at the same rate that it leaves. No deflation.
Anyway, I've seen rolls of bubble wrap on the shelves sit around for many years. They don't just deflate.
OTOH, foamboard does actually deflate over time. Published R values for foamboard are based on testing the board as it comes out of the factory. As the "blowing agent" (the gas within the bubbles in closed cell foam) diffuses out through the foam and is replaced by air, the R value decreases. It takes a few years for this to happen, but it does happen well within the lifetime of the structure.
Still, I expect that the R per inch of foamboard is much higher than the R per inch of bubble wrap, but it is an interesting question as to how much.
Bubble wrap is probably awful in a fire, but exposed foam is probably worse.