There are two other possibilities for this type problem, and two solutions:
the ignition lock/key assembly fits into the cast housing behind the "security cage" on the steering column. Even the newest SVX's are a dozen years old, now. On my car the key could even be removed while the car was running
. At the other end of this cast housing is an ignition switch & pigtail that can look good from the outside but cause the exact symptoms described.
I replaced both. It is a good idea to replace the headless "security" screws/bolts with regular bolts or to cut a flat blade screwdriver slot in these fasteners to facilitate removal and tightening. Mine were always coming loose (easily determined when the entire ignition lock assembly rattles and moves around).
I was told I "needed a new starter" for about 10 years, from 3 different shops: my original starter works wonderfully and has for 17 years and almost 175,000 miles.
Cheers!