You're right that as a person moves into management/staff+ that individual details matter less for them, because they begin to deal at a more strategic or operational level, but someone still needs to obsess over the small details (to make things easier, let's assume we're only talking about the details that actually matter and that there's broad agreement what that set is). It's not that those details stop mattering to them, it's that they have to depend on the assumption/promise that someone else is going to mind those details for them, so that they can work on things with a broader perspective.

The problem comes when no one is actually minding those details anymore. If those details aren't actually minded, then the work at the higher levels will be much less useful and effective, and in the worst cases won't matter at all.