My recollection of history was that Einstein's famous equation is the fundamental idea that got people thinking about the bomb. He is also credited with writing a very influential letter to Roosevelt arguing that the Nazi's were working on the bomb and the US should go full steam ahead to develpe the bomb. It has been reported that this letter was very important in FDR going ahead with the Manhatten project. You are correct that Eienstein had little to do with the practical engineering of the design of the bomb.
This is more a function of the public's infatuation with Einstein than a key component of the bomb's development. A number of people independently realized the possibility of a chain reaction. They pointed it out to Einstein who pointed it out to FDR. That was the extent of his involvement. It would have been developed without him. Now would we have figured out time dilation until we tried to build a GPS system? I dunno.