"And as the radar dish sweeps around, shining its beam on regions untouched by magnetic field, is this not the same thing as "turning on the field" there?"
No, I'd say that analogy simply obfuscates the issue. We already know the speed at which electromagnetic waves propagate. Those electromagnetic waves propagate at the same speed as do disturbances inside an existing magnetic field.
On the other hand, what we are debating is not how fast those disturbances or electromagnetic waves propagate, but rather how fast the magnetic field itself propagates when it is first formed, as well as when it is turned off.
Waves can be found *inside* the field, but waves are not the field itself. Waves are known travel at a measurable and accepted speed, something that is still under debate for the field itself.
On the other hand, what we are debating is not how fast those disturbances or electromagnetic waves propagate, but rather how fast the magnetic field itself propagates when it is first formed, as well as when it is turned off. OK, so what does it mean to "turn on" or "turn off" a gravitational field? Physically, how do you do it? How long has it been since we've had to worry about the gravitational field from the sun "turning on"?
And since all charge is conserved, what does it mean to "turn on" an electric field? Oh, you can charge up a capacitor, sure, but you're simply moving existing electric fields into a new orientation. The only thing you're ever doing is changing and reorienting the existing field, so what exactly needs to propagate, here?
And since all magnetic fields are contingent upon point of view, what does it mean to "turn on" a magnetic field?Every charge is moving from somebody's point of view. Furthermore, even electrons have to change velocities continuously: the accelerations they undergo may be large, but they aren't infinite, so all changes to any magnetic field must ultimately be continuous. There's no such thing as a truly "sudden" turn-on for a magnetic field; it's always gradual on some timescale.