This is an important question. Two quick interrelated responses come to mind:
1. Teams try to make them into "normal" QBs at least part of the time because no NFL team can win consistently without a QB who, whatever his other skills, is also able to stand in the pocket and deliver an accurate pass. Mobility is great, but without that skill, a guy cannot be a consistently successful starter, because the better teams will keep such a QB in the pocket and dare him to beat them with accurate passing.
2. Related to that, no player paid what VY is paid can be used as a part-timer. After all, someone else would have to play "normal" QB, which would mean either you have two guys getting paid starting QB money or you have a bargain basement QB taking a large proportion of your snaps. That makes no financial or strategic sense. It is also an open question whether VY would be willing to accept such a role. We know that Kordell chafed under the assumption that he was just a novelty act. He wanted to take all the snaps at QB, and the Steelers decided they had to try it, with the results that we all saw--intermittent success and inability to beat top teams when it counted.