I assume most people are assuming a web/mobile startup on this thread - I'm thinking along those lines at least. For other areas, it's different. A long time ago, I worked for a CTO at a silicon chip startup for example; he had a very different role.
CTOs can do a lot, but I think there's a lot of flexibility in how you define the role depending on who his or her team members are. So, for example, I don't doubt Karl was the Social Media Manager and held the CTO title while he did it, but I'm not sure how helpful it is to highlight that specifically... it really just highlights that when you don't have a social media manager yet and you need one, you have to rely on adaptable people like Karl. His ability to do that makes him a great startup team member, no doubt, but I don't think it makes anyone a better or worse CTO to be able to step up like that.
It might be more interesting to pose the question of what uniquely should a CTO not mess up. In other words, what are some things that only he or she can tank the company on. It's kind of like that line about the Tour de France: you can't win it today, but you sure can lose it.
For me, it's:
1) Technical interviews of your engineering staff (regardless of who ends up managing them and regardless of who conducts the interview)
2) Where do you draw the internal lines between your systems. (The problem with this one is if you do this right, no one will notice you did anything at all.)
That's pretty much it for me. After that, it's great if he or she can be a good startup citizen and work like crazy on whatever they can do - code, manage, recruit, social media I guess... but that's all about contributing, not uniquely contributing - other roles in the company can potentially contribute to those areas too.
Note that both of these are expertise requirements, not hustle requirements. You should pay attention to the startup CTO who seems to not work that hard, yet is never surrounded by drama; that's not necessarily a bad sign. In contrast, you might want to be at least a little suspicious of the CTO who is consistently working 80 hour weeks and having the servers crash constantly. At a minimum, you are severely under-staffing him or her: that's nothing to brag on.
If you have a CTO and do not have a different head of engineering management, then they also need to fulfill the requirements that only a VP of Eng can mess up:
1) Sourcing of technical candidates at a volume high enough that you can make on average one hire per month (or more if you're growing fast - but even if you're not, you need to be able to close technical reqs in a reasonable time span or people start to hate the hiring process - 4 weeks is a good goal).
2) An organizational system of updating and QAing the code base that can still function reasonably well even if the head of engineering is away for 2 weeks.
3) An engineering culture that is neither overly-argumentative nor suspicious of new hires (i.e. inclusive, not exclusive).
4) A product+engineering culture that has product people and engineering people feeling like teammates.
Anything after achieving all that is again bonus in my mind. In other words, if you have that, then increasing your productivity should be a question of spending more money. If you're cash constrained, you might want the VP of Engineering to also be coding, for example, to add to your productivity without spending more. However, if you don't have the above, then adding money generally just creates a lot of drama, not more output. Note that none of the requirements in theory require technical expertise, but in practice it's unrealistic to think you'll achieve #2 or #3 without it.
If your CTO is the head of engineering and also the head of product, then I'd argue he or she better also be a co-founder... And once your title is co-founder, I'd argue it doesn't matter what else your title also is: you are responsible for cash flow and hiring (full cycle: sourcing, screening, closing, retaining). If you do that well, nothing else that you do really matters (I mean, you'll work hard and you'll feel like it matters, but you're probably in good shape regardless). If you do that poorly, nothing else that you do will make up for that failure unfortunately.