1996 Buick Skylark car won't start

the car won't start. when the key is on all dash lights come on, i got headlights, radio works. when i turn the key to the start position, everything dims, but starter doesn't turn over the engine. i get no noise from the solenoid, but when i use a screwdriver and jump across the two terminals on the solenoid it started. i checked all the fuses and they are good.battery is good.
can the neutral safety switch keep it from starting if it has gone bad?
how about the gear selector switch. the wiring schematics show the wire coming from the keyswitch going to the gear selector switch then to the starter.