In a typical flight cycle of a modern jet, operating from typical runways, the OEI takeoff case is NOT the most demanding operating point. Top of climb calls for more "oomph" than takeoff. This assertion is supported by the fact that a great majority of takeoffs are made either derated, or flex (ATM), or some combination of the two.
But going back to JATO - I once worked around a DC-3 fleet with JATO installed. They weren't fired until after V1.