One thing I forgot to mention...
Lets see if I can remember this as its been a while. In order for an aircraft to climb, the thrust must exceed the drag. Now shortly after take off the aircraft has the gear and possibly some flap out also, the drag is very high. Suddenly the thrust is reduced by half and the gear and possibly flaps are still extended. Additionally if you are unliky enough the engine is not developing partial power but a complete engine failure exists so control forces are required further increasing drag whilst you try and pull through the air a spinning flat disk attached to the wing (engine and propeller). Normally, there is now insufficient thrust to even maintain level flight with the gear and maybe flaps out. Obviously the only thing the pilot can do is lower the nose. But this all happens very quickly, the pilot tries to keep the nose straight and then remembers to lower to nose. So to even contemplate attempting a climb the pilot must:
1. Recognise that an engine failure has occured
2. Clean up the aircraft and Identify the failed engine
3. Feather the failed engines propeller
Only then, a climb maybe possible after taking into consideration many other factors beyond the pilots control.