IIRC what a solenoid does is act like a big relay for the starter. The starter signal energizes a coil which jumps the power wire from the battery to the starter motor. When you use a screwdriver to start the car you bypass the solenoid.
Since you’re not hearing the solenoid coil engage I am assuming you have a bad solenoid coil this not allowing power from the battery to the starter motor. If you can hear the clicking but the starter motor isn’t turning you could have a dirty plunger and not allowing power to go to the starter motor or not enough amps/voltage coming from the battery.
Sent from my iPhone using Tapatalk
__________________
¯\_(ツ)_/¯
|