I'd say it's important for a CTO to be a CTO. I've been at too many places where the CTO only wanted to be a software developer/architect and would not focus on process, cost, employees, etc. CTO is a leadership role, and leadership means you have to take care of all the responsibilities under your authority, not just the interesting ones.
IMHO, too many people chase the CTO title but few really want the actual job.
IMHO, too many people chase the CTO title but few really want the actual job.