The takeoff flight profile is dictated by the takeoff flight path certification requirements of F.A.R. Part 25 (or its equivalent). The carrier can modify the regulatory takeoff flight path segments, but only with the concurrence of the aircraft O.E.M and usually also with the concurrence of the state of registration aviation authority.
The crew is most decidedly not competent to further modify that profile in the event of an engine failure on takeoff. They have no idea of the location of critical obstacles, clearance of which may only be assured by following the takeoff flight path with precision.
Further, failure to follow the flap retraction schedule with OEI, especially with a heavy aircraft at a critical airport, can result in running out of airspeed, altitude, and ideas, all at the same time.