Why don't motors coupled to gearboxes spike up significantly?
Your expectations are wrong. The "4-7 times" statement is what is known as a "rule of thumb", used for rough estimation. It doesn't mean that every motor ever built draws a minimum of 4&times; its rated current at start-up.

In your situation, it would seem that the motor is able to spin up fairly quickly, limiting the time duration of the current "spike". Plus, the inductance of the windings limits the rise time of any spike. Your meter may also have a limited response time. Together, these effects limit the height of the spike to what you see.