Sometimes the visionary entrepreneur (you?) is not the best to manage the company or part of it. And if this is you, you know it and don't even like it.
The COO can become an integrated part of your team and take personal responsibility for managing your company and leading change. Keeping everyone accountable and on target is vital to get your vision done.
There is a difference between building a business and building a company. The latter is less important initially and will become more critical as the company grows. Taking care of the company is the core job of the COO. If the job is undone, you'll undermine your company's ability to execute your vision.
I like to see the company as the product the COO builds and develops. Whatever the method, the goal is to increase the company's strategic ability and business and employee value. It's also a prioritisation exercise, as fixing every part of the company is not realistic or sensible.
The COO should also be an expert coach and facilitator. The job of building and developing the company always leads to change. The skillset helps the COO guide individuals and teams to focus on things that matter the most and helps reach sustainable change.
Which jobs should the COO do in your company? It depends on your situation.
Sometimes being your coach is enough. Sometimes becoming your team's coach is warranted. And sometimes full-on management responsibility is the only way to go.
A competent COO knows which scope is justified for the desired result.