The idea behind the FO doing all starts is that the CAPT can focus "outside" and manage ground movements.
As the captain has the entire responsibility for the safe conduct of the flight by law, that would include observing the correct procedure for starting all the engines.
If, as you say, the captain stays focussing "outside" and managing ground movements, then who is double checking if the first officer is starting and monitoring the parameters of each engine correctly?
Depending on individual company procedure, some have the captain responsible for the engine starting and another company may have the co-pilot responsible. Either way, both pilots should be closely observing each engine start especially as a defective start could lead to serious engine damage - for example over temperature. It can take only a few seconds to ruin an engine costing multi million dollars which is why it is vital both pilots watch the whole start procedure until the engines are stabilised. That includes a push-back start.