Many people hold their cutting head steady with a finger during the scoring process. This is not necessary.
The axel of the cutting wheel is slightly forward of the centre line of the cutter. In addition, the cutter is held slightly angled back toward the operator to be able to see the wheel and the cartoon (or marker) line. Both of these act to ensure the wheel follows the movement of the arm or body in a forward motion.
In cycling, the distance between the angle of the shaft of the cutter and the axel is called the “trail”. The greater the amount of trail, the easier it is to keep the bicycle following a straight line. The same applies to the cutter. This trail is created by the extension of the angle of the cutter to the glass. The axel of the wheel is behind that line. The cutting head has a sharper angle at the back than the front to accommodate this angle backwards. The resultant forward force is in front of the axel and so leads the wheel to follow the direction of the cutter without any need for stabilisation.
There is no need to have your finger on the cutting head. It swivels for a reason. It will follow the direction you are pushing without any angle, so there is a clean score. If you attempt to stabilise the cutter head you risk the wheel running at a slight angle to the direction of the score. I talk about this as a skidding score. The result of this is to give a score with forces directed not only straight down but sideways too. This gives the glass many more ways to break. And not always along the line you want.
Also when you want to score a tight curve, the slight movement of the head allows the curve to be slightly smoothed again without any skidding. This means there will be fewer pressure lines sideways to the score line.
Manufacturers have put the play into the cutter heads for a reason. The above attempts to explain it. The manufacturers would not include a feature that costs time and effort, as well as cost if it had no purpose. It seems perverse of us to try to run counter to that by holding the head or even fixing it solid, so it is unable to pivot at all.