Don't be so quick to blame the starter just yet. ( it might be bad, we don't know just yet )

Make sure the wire that triggers the solenoid is capable of passing high current. Make sure the circuit is good through any neutral safety / anti theft systems, to the ignition switch then back to the battery + from the ignition switch.

There are two windings in a starter solenoid that pull in the drive / switches battery power to the motor, here is how it works.

The first is a pull in winding, this is a high power / intermittent duty winding that gives the drive a initial tug towards the flywheel. One end of this winding is tied to a positive solenoid wire, the other is tied _to the positive side of the motor_ . ( remember this, it is a concept that is used later )

The second is a hold in winding, this is a low power / near continuous duty winding that keeps the drive pulled in. One end of the winding is tied to the solenoid wire, the other is tied _ to ground _.

So, the sequence of events is:

Positive battery power is applied to the solenoid wire,

As a separate action from below, the hold in coil is energized.

The starter motor starts to rotate slowly because it is in series with the pull in winding,

At the same time the drive is headed towards the flywheel ( this slow rotation assists in meshing of the gears )

When the solenoid bottoms out and the drive is fully shifted, a contact disc bridges battery + to the motor causing it to rotate at cranking speed. The electrical engineering guys will see that the pull in coil now has + voltage on both sides effectively taking it out of the circuit.

Do a search for " hot start kit " , air cooled VW / Porsche and oval track guys use a remote mount Ford type solenoid as a fix for a weak starter solenoid circuit.

As a quick test you can rig up a 12 Gauge wire from the battery through a heavy duty push button switch ( like 50 amp ish ) to the starter solenoid. This bypasses all of the car cranking wiring. Be careful, the neutral interlock is now bypassed and the car can start in gear and drive away.