The role of the CTO changes with the size and type of the company. The CTO of IBM is a person with a completely differrent skill set than the CTO of a startup.
When your funding comes from Friends, family and yourself, your CTO should be the person who can implement your product him/herself along with some management skills to guide a few developers. THis gives you an option to have him/her build the whole system, build critical parts of the system and outsource everything else, or outsource everything and have him/her manage the outsourced work. Generally, at this stage, a 2-pizza team is sufficient, and you need the CTO to be able to manage this team. At this point, you don't really want to hire a dedicated manager.
Once you grow to 50 people, you need the CTO to know how to hire the right mix of people to make this organization function. At this level, being able to create a sustainable R&D organization is all about setting the right culture and hiring the right people. This person will also probably need to be more outward facing. S/he will probably be the technical face of your company, and will need to be able to provide credibility to your organization. Also, at this point, this person will also need to focus more on the comptetion and the market than on the day to day R&D
Once you get to 500 people, then your CTO is more or less a people manager. He would still need to technical know how to understand the problem, and propose solutions and provide strategy. However, s/he would be really relying on other people to do the leg work, and mostly providing direction. This kind of person is a very differrent person than the one you need when you were a 5 person company. This CTO is more of a "tell other people what to do" kind of person than a "get things done" kind of person.