Steve I agree we can not tap the range outlet for a 20 amp outlet
I do want to comment on this.
Using the third wire on a 3 wire range circuit for the neutral is a no no.
The wire you are talking about is
a neutral it is not a grounding conductor, the stove or dryer uses this conductor to run the 120 volt loads, motors, lamps etc.
The code did permit the neutral to be used as EGC, it did not allow the EGC to be used as a neutral.