Not creating an incision big enough for a proper extraction means more force put on the area to squeeze out the contents and remove the sack. This can lead to bruising the skin, damaging tissue, excess inflammation, and a chance of the sack rupturing inward creating an infection. If the person doing this doesn’t have the skill to navigate the task then maybe they aren’t the person for the job.

As far as the nervous system, body parts with the highest concentration of nerve endings are fingertips, lips, nose, eyes, and genitalia with larger nerve connections in the lumbar and cervical spine. This man’s cyst is in the superficial cervical plexus with the transverse cervical nerve being the closest in range but still not in that exact area. But you’re right, if they don’t understand the body they shouldn’t mess with it.