I've only experienced one incipient spin from a student. In this case, a power-on stall, the problem wasn't the failure to keep coordinated during the set up, but rather to maintain coordination when the stall broke. He failed to release the pressure and we dipped heavily to the right. Fortunately, I saw it coming and recovered after only a 1/4 turn.
I think you are right, "keep the ball centered" doesn't cut it. The student should be able to do the stalls and slow flight with the turn coordinator covered. I think we give the ball way too much significance. Like all visual flying, coordination should be maintained by looking out the window.
In the straight ahead versions of anything, including stalls, coordination simply means that when the wings are level, the nose isn't moving sideways (yawing). Period. The secret is to teach the student to recognize that movement visually (with butt-feel also, but since we're usually sitting on the CG I think that most of us don't really feel yaw until it gets really bad.
Once we get used to seeing that movement, it can be used even in turning maneuvers.