Some pitchers have been known to mix it up.
Some coaches teach “righties on the right, lefties on the left” to create angle. I suggest you consider this: If you could move the full length of the rubber, your release point would move over 18". If you drew a line from the catcher’s mitt to each release point and looked at the angle between them where the lines cross home plate, how much difference is there really? Do you think it is significant to good hitters? Just something to think about.
The NPA teaches positioning yourself based on your drag line to minimize the chances for a late posture change. For example, if you’re a RHP who strides closed, starting on the right side of the rubber takes you further offline from the target and that increases your chances of shifting your posture as your shoulders square up to the target. Such a posture change can affect control, it can pull your release point back and raise it up which will affect movement and it can expose you to greater chances for injury.