Phosphorus is a component of the ashes, but potassium and calcium are also major components (depends on the type of wood to a certain degree).
The discrepancy between those getting dead spots from applying ashes versus those who don't probably has a lot to do with the native pH of their soils. Those with naturally low-pH soils won't have a problem with too much alkalinity from the ashes, but those with already high pH soils can.
Ashes are good to use at the same time you're applying natural nitrogen sources (i.e. composted manure). Since natural nitrogen sources usually contain most of their nitrogen in the form of amines (lot of N-H bonds), when they break down to form nitrates (N-O bonds) they release hydrogen ions into the soil. Since those are acidic, it is good to have the basic ashes around to keep the soil pH nearly neutral. Along these same lines, if you have a lot of oak trees around, applying ashes to your lawn can balance out the acidity that leaches out of the oak leaves in the fall.