If I was Captain again, and taxi lines were such that I think we could be OFF in 5-10 min, and I was flying with clone me in the right seat, I'd do it this way:

Push, awesome clone me starts left engine, I enter the data, they disconnect, they waive, we do the after start and go. As soon as the coast is clear start the right engine. Then do taxi checklist then before and go. Unless I want to do something else. If the numbers come late, awesome clone me taxis and I enter.

That said, real world me, realizes you will have a better FO if you keep them in their comfort zone and routine. Do that and less mistakes will be made. There may be a better way to do things, like my way, but it might not be the cultural norm the FO is used to and the FO has their show to run too.

So, bucking that norm isn't going to do the FO and thus me any favors. So I'd ask the FO, how do you like to do things? Get a feel, game plan from there, call audibles from there.

And just because atc clears you or you see no traffic and a wide open chance to go, you don't have to go. Some people have get there itis despite being leg 3 of day 2 of 4 days. Relax.