Concerning lateral pterygoid muscle,why protrusive action functions to test intactness of mandibular nerve?

Not really: The lateral pterygoid muscle is the only muscle to open the mandible jaw while there are many muscle group to close the jaw. Protrusive action is one of movement possible when you open the jaw. While Mandibular nerve is a sensory and motor nerve from cranial Trigeminal nerve. If impair happened on this nerve, then it will affect the movement of the mandible jaw whether in open or close motion.

Lateral pterygoid: The motor function of the mandibular branch of the fifth cranial nerve is tested by asking the patient to clench their teeth together and to open their mouth against resistance. The "thumb" test is the only reliable test to examine lateral pterygoid muscle.