In-universe: The Mirandas were probably cheaper to produce. And they seem to have been easily adaptable to a variety of roles. We see them with the roll-bar (TWoK), without (USS Lantree in TNG), and with other things glued onto the hull (USS Bozeman). Also, the Constitution class ships were (I assume) mostly sent out on long-range/long-duration missions, and there were probably fewer of them left to last as long as the Mirandas seem to have.
Real-world: TPTB were probably afraid that the casual viewer would confuse any Constitution class ship with Enterprise and think that Kirk was going to show up any second.