I also removed the extra Floor and added a Clamp to stop negative numbers from affecting the emissive.
After making the burn effect, I realized it was similar to a great effect I saw in a GDC presentation about Substance Designer. Here's what I came up with for that, though I imagine there are better ways to achieve it.
(any of the gradients can be replaced with a Levels adjustment so if you don't want the extra utility of offsetting the age your graph can just be a Blend and a Levels)