SPC monitoring is necessary for customer-specific requirements and products/processes at a high-risk level. You monitor SPC based on customer requirements for the given product/process.
If you are referring to the Normal distribution and the relation between the distribution and the P-value, please note that when P-value is lower than 0.05 distribution is Normal.
On the other hand, if you are referring to linearity, linearity is a relation between X and Y which can be, for example, positive or negative.
The relation between a normal distribution and SPC can be explained as follows:
When Y has a normal distribution, it means that it follows a central tendency. It also means, based on the theory of central tendency, that it has a variation that we can use to predict the values of our Y.
A normal process has special causes that are easy to spot on the SPC chart when they occur. We cannot predict with certainty a Y that does not have a normal distribution.
That is why we always strive to identify and eliminate special causes and to identify and reduce the impact of common causes so that our process would be as normal (pointy) as possible.