That has a chance of working due the very fast rise time you might be able to induce a sufficiently large voltage to cause some actual damage, it will be a 'one shot' affair though. But anything involving regular coils and some push button arrangement will likely cause it's own insulation to break down long before it will do damage to a phone.
That's one of the reasons the spark gap is there in the first place, without it you'd hit the limits of the coil insulation very quickly.
Spark gaps are a pretty nasty way to generate EM interference so there is some truth to the story.
But even with my plasma cutter (12.5 KW, so not the smallest in the catalog) and stick welder I never managed to crash a cell phone.
The plasmacutter did have the habit of resetting the controller until I properly decoupled all the lines into it.