"It is a neat idea, oddly not used by the real rocket boys, who when push comes to shove favor simplicity over efficiency. "
Two issues, one already mentioned (drag/weight penalty once a certain flight speed is reached), the second is that at lift-off you have potential for reversed exhaust flow up the side of your rocket...you know, where the nasty volatile stuff you don't want to burn yet is stored...that potential exists because typical booster engines operate below stoichiometric oxidizer/fuel ratios (aka fuel rich), as these ratios maximize thrust. But, rich mixtures also re-ignite in the sea-level atmosphere...and if they re-ignite before any vehicle speed has developed (which they will), there is a good chance for the reversed flow. The above may or may not hold for solid propellants, as I'm not sure of the exhaust species for a typical solid grain, but I'm reasonably confident it would hold for typical liquid boosters.
At a certain rocket company, they did look at whether that low-altitude exhaust energy could be recovered by some type of eductor (with a rig to maintain pressure via external blown air until liftoff was achieved), or even a turbine-driven fan to maintain positive pressure, but the concept quickly got heavy and complex. And, the analysis carried the assumption that the eductor's weight could be jettisoned at an optimal point in the flight, not necessarily a simple thing to do with a running booster engine still dumping exhaust through the dang thing.
Rocket engine designs are horribly complex, and every effort is made to squeeze extra ergs of energy out of the fuel carried whenever possible...if the mission analysis says it's worth it. You're trading extra pounds of fuel to be carried vs. extra lbs. of engine weight. Fuel gets burned, so the weight penalty decays with flight duration, but engine weight stays until you stage/seperate.
File this under the "more than you wanted to know" column.
