Being a founder is about responsibility
When you start a company alone, you are 100% responsible for every job it entails. You are a true generalist. This means your job consists of product strategy, customer service, development/engineering, UX design, finance, accounting, janitorial work, hiring, distribution, etc. etc.
As you bring new people into the organization, they slowly start taking responsibility from you, job by job. So you hire an engineer, and now they are responsible for 90% of product development while you take the remaining 10%. The talent base has grown but you never abdicate all responsibility for a job... you will always play at least a small role in everything.
You keep hiring people until most of the job responsibilities rest on other people. When this happens, you still play a role in everything, but it is a role more of vision, guidance and support.
So the office is dirty and nobody's cleaning it up? Yeah, it's not your primary concern -- but you're still responsible for it. "That's not my job" shouldn't be in your vocabulary whether you have 10 people working for you or 10,000.
At Apple, they assign DRIs (directly responsible individuals) for projects and tasks that need getting done. This is the go-to person that takes ultimate responsibility for the job. It is important that someone is assigned this role because otherwise responsibility may be too diffused, in which case nothing gets done.
So in my view even though the higher up the chain you are the more your responsibility, there still needs to be a specific DRI assigned for each project or job in the business. A DRI is not used to assign blame, but to assign responsibility for maintenance, improvement, and problem resolution. This could be a manager or just someone on a team that came up with a good idea.
Cross posted at the Atlastory Blog