What would cause the plunger to not go in all the way?
The plunger stuff is very unlikely but could have gotten bent, small screw could have fallen down inside, etc. I assume when you have the starter off and touch 12v to the solenoid terminal with the starter body grounded that the bendix gear shoots out violently? Ok good.
Note: clutch pinion gear = old school bendix gear
1) With the starter on the cruiser I would try taking the stock ignition wire off the solenoid and run a new piece of wire up by the battery. Have someone turn the ignition key to the start position and touch the wire to the positive battery terminal. If it turns over and starts make sure they quickly release the key to the run position.
If it still just clicks then step 2. I would very carefully run one jumper cable down to the big lug on the starter side of the solenoid (the opposite side of where the big wire comes down from the battery – pull back rubber boot). Be very careful that the clamp only clamps on the terminal and does not touch the starter or anything else. Either with the stock wire hooked back up or touching the small wire you ran to the solenoid get the solenoid to click and then hit (touch) the other end of the jumper cable to the positive battery terminal. Use common sense and don’t start a fire.
Hopefully what 1) should eliminate is any problem with power to the solenoid dropping out after the key is turned.
Just doing 2) assuming a strong battery 12 volts should get the starter motor to spin without power going through the solenoid. By having the solenoid click first and assuming the bendix gear is shooting out, the cruiser engine should spin.
Somehow I can’t believe you are making a mistake with the contacts so the above may be futile. If you are inclined to save some money or are curious I would pull the starter motor apart and look at the brushes. There are continuity tests for the armature or if you know someone with a growler (not Junk’s sister) they can test it for you. I used to turn the commutators down on a lathe for a smooth surface for new brushes to ride on and scratch in between the commutator bars to make sure they were not shorted at that point. It is possible one of the wires in the fields is broken and not always making contact or a field is shorting to the starter body intermittently. Again continuity tests.
Most don’t want to do the testing or have the time and just replace the starter.
Btw, when you have the starter off look in at the flex plate at the ring gear and see if there are any burrs and try to jiggle the gear. Sometimes the gear can bust or crack and jam so the engine can’t turn over at that point on the gear.
Sorry for the long post.
Bill