Centaurus, c100driver is correct about the captain's input.
Furthermore if FO ALWAYS would start engines then captain can concentrate fully on pushback, monitor FO and possibly use hand signals without having hands on engine start levers etc.
Personally, the philosophy of handing over controls in some LCC's is too complicated, handing over control before taxi introduces this. Simply take LHS/RHS responsibilities from before engine start/push back and thus as per Boeing FCOM Master document and it would be a lot easier.