You have the ultimate in a simple diagnostic problem. You do have a symptom and since it is predictable, what else do you need.
A problem exists when you don't have a repeatable symptom. Your problem is easily testable because it exists. Don't look for it when it doesn't exist. You have two circuits to verify. The first is the basic power circuit to the starter (the large positive battery terminal). The test is simple; it must stay the same as battery voltage all the way to the starter. If its is 12v at the center of the battery post and 12v at the large connector on the starter, it has done its job. It must pass this test while the problem exists!!!
The second test is the activation circuit (#50 on the wiring diagram). This circuit should be the same voltage as the battery when the starting attempt is made.
A third observation can also help; what is the change in battery voltage while an attempt is being made. If the battery voltage stays high and both power and term# 50 are the same during the starting attempt, then an internal OPEN circuit exists in the solenoid or its grounding circuit through the armature. If the battery voltage drops then a load has been applied. This would be a short. If the voltage drops below 9v without any movement you have a dead battery (or a huge wire burning short - shorts on power circuits, burn wires; unless they burn a fuse).
There are a few switches involved that shouldn't be temp dependant, but when the problem exists it is the most basic of circuits. You will learn to evaluate this circuit or you should forget dealing with such.
Bosch Master, ASE Master, L1
33 years MB technician