As a coach I believe there are two important things that should happen even before the coaching session begins.
First is the coach should ask the student what he hopes to achieve (more consistency, improved break-building, turn pro?) and then the coach should explain what is achievable through coaching and also explain improvement takes HARD SOLO PRACTICE and just playing one or two frames a week in a league won't get a player anywhere, even with the best coaching in the world.
Secondly, and this is for players who have played awhile and achieved some level of skill, is the coach must ask the player what the PLAYER believes are his problems. As has been said when a player comes to a coach sometimes they tend to be on their 'best behaviour' and it's very difficult for the coach to detect anything badly wrong.
I have also found that video analysis does reveal EVERY little fault in a player's technique and it's also great to actually show the student and explain in real time what the problem is and showing how it effects the straight delivery of the cue. I should warn players though, when using Kinovea for instance in the frame-by-frame mode you will see a lot of problems and some of which are important and will effect the delivery but others might be the player's own natural glitches which he is either used to or else must use because of his individual physique. Be very careful in trying to iron out everything you see.