Let me start with my lessons learned and save you few years of trial and error experience - Successful managers pay attention to the small details.
As an important large-scale manager, you cannot run or shy away away from paying attention to the small details.
This is something that was a common thread in different practices I had - Product Delivery, Product Management and Sales.
Small details are extremely important and the success or failure of the venture, no matter if it a sale-cycle or product design. When you are responsible for the execution of that venture, even as a hotshot manager, you must pay attention to these details, in order to execute.
I've seen this with successful managers and less successful ones ( and I thank each one of them for that important lesson). The successful ones were trying to understand the details, the difficulties, to make sure they can get into the depth of the problem or the discussion.
It can be understanding every line-item in a pricing spread sheet discussed with the customer, reviewing the workflow of a new feature design or asking difficult question when a product development plan is presented to you. Not taking anything for granted and not thinking that this is "beneath them".
Obviously focus is extremely important, and the manager shouldn't do everyone's job. BUT... when problems happens (and they usually do) and big-manager's-decisions has to be made, then the dive-in needs to happen.
Understand this, if the issues or discussions are less important to you and your focus, they are probably important to some other stakeholder, which makes them eventually important to you as well, the process leader.
If your end goal is "getting it done" and the stakeholders are the people that help you getting it done, you (should) understand that the small details will lead you to these big wins.