Where is #1?

Before almost every game, a team will take a round of infield/outfield as a tune-up. It's a chance to get the blood flowing, get a feel for how the ball is going to bounce, and perhaps practice seeing the ball against a tough sky. There is a key player often absent from this routine...the pitcher. During the course of the game, the pitcher is often called on to field a groundball. The pitcher is required to make some very tough throws at times as well (e.g. starting a double play up the middle OR throwing to a moving 2nd baseman covering 1st on a bunt).

Are we missing something here? Why is the pitcher traditionally left out?