A couple of thoughts.
First, when you mash the starter pedal part of the way down with your foot, that action drives the linkage against the yoke lever that comes out of the top of the starter motor.
As you mash the pedal on down, the linkage engages the yoke lever. The yoke lever moves the starter motor gear against the flywheel.
Simultaneously, the linkage causes the top of the yoke lever to press against the button on the electric switch (the one your son is pressing with his fingers, pretending to be the yoke lever) on top of the starter motor.
Mashin' (or stomin') on the starter in the cab is supposed to do all of this for you, but your's has a hitch in the git-a-long.
Any slop in the linkage from the foot stomper to the yoke lever will make it more difficult to overcome the return-spring tension in that spring you see just ahead of the yoke lever. Nevertheless, you should be able to overcome that, and it sounds like you can by stompin' down as hard as you do on the foot pedal. If the linkage bars from the foot pedal are not necked down, the holes not wallered out, and the linkage connections tight, you should not have any slop in the linkage.
If you don't have much slop in the linkage, the easy thing to look for (after you have disconnected the battery cable from the battery) is to take the electric switch off the top of the starter motor and check it for wear. The switch is just a moveable contact bar from the battery stud to the electricl stud coming up out of the starter motor.
These things get worn. If there is wear, try turning the moveable contact bar 180-degrees. If the stud contactor coming up out of the starter motor is badly worn, you may have to do a replacement, or replace the starter motor.
Putting some kind of a shim on the button may be a temporary fix, but, as has been said, if you want to fix it up right, find out where the wear is, get the linkage all tight and tidy and replace the electric switch on top of the starter motor.
That is a fairly easy job, and you should have a nice working foot-stomper system. It will keep your son happy by never havin' to git his fingers mashed jest to git dad's truck cranked up.
