I think you are right that Bill does talk about mechanical issues that pull the shoulders out of line for left or right misses. I think this would explain misses but for a consistently off the mark to one side is more what I was referring to.I am in no way a pitching coach.
However I recall Bill Hillhouse stating that missing up or down is a timing issue and left and right is a mechanics issue.
Every pitchers issues are different but my DD does something very similar to the OP. When she starts to warm up at full pitch, the first couple of pitches she consistently misses to the arm side of the target which for her is the the right. In about 10-20 pitches she slowly reels it in to the target until she is locked in. So in warming up she is first establishing her timing and natural IR release for that day and not so much worrying about direction. After she establishes her release timing she is adjusting her aim until she locks it in and then she is good to go.