In my experience, no they do not. They're waaaay too busy. CEOs are a different story entirely depending on what the sales/awareness model of the business is. CEOs go out and Tell The Story to any industry stakeholder that will listen. CTOs post up with their team and get shit done. Thats my experience and what I saw from my peers in other startups.

I don't see how the need for business travel would arise. I perceive the CTO role as being responsible for ramping up infrastructure and managing the development team.

In my experience, no. They are cranking out code and managing directs (engineering, data, infra) 8-12 hours a day, 5-6 days a week.