They have to distinguish between PS3 and PK4, because they can't have 3 times as many PK4 kids - that would be a nightmare when transitioning into K.
The year my child was in PK4 at Stokes, he was only one of 5 children moving from PK4 into K that year. So the next year- there were more PK-4s and not many PS-3 spots that year. Coincidentally, there were many children that got spots in K that year. So, although your theory makes sense, I'm not sure they can do it exactly balanced between 3 yo and 4 yo.