I think the type of chest youre on about is not just a deep chest associated with a running dog, but also the width of the chest, if a dog has a wide ribcage aswell as a deep one it will have a greater lung capacity than a dog with a deep but narrow chest. Bigger lung capacity= more efficient delivery of oxygen to the working muscles therefore greater stamina potential.
Look at alot of whippets built for short bursts, they have deep but narrow chests, a dog that is not renound for its stamina but for pure explosive speed, also comming in greyhounds, although many coursing bred greyhounds