This discussion of organs strikes me as a bit strange.
First - Church organs are often pipe organs that sound (to me at least) dramatically different than Hammond electric organs frequently played by jazz musicians.Though there are other electric organs also used in jazz and other settings.
Second - Jazz organists do not all sound the same. Compare for example Jimmy Smith with Wild Bill Davis. Also think of Larry Young, Don Patton, Milt Buckner, Akiko, Shirley Scott, Mike LeDonne, Johnny Hammond Smith, Lonnie Smith,Bill Doggett, and on and on.
So to refer to jazz organists as sounding like Roller Rink Roly Poly organists is, in my view far far from accurate.
Sorry, Don Patterson Not Patton.