I'll give this a shot.
I'm interning at a small non-profit in right now and the type of job you're describing is exactly what the people in my office do. Majority of the time is spent on office work, but they also travel frequently to oversee their international projects. Average duration of travel time at any given point for one employee is proably one month per trip and I think they travel on big assignments like that two or three times a year. From what I can tell from this office, having an MPH doesn't give you an advantage over someone with another international development type degree.
My office mates who are technical officers and program officers have MPHs, MPPs, MPAs, MAs and I am sure there are a few others that I have missed. I think as long as the content of your degree is in line with want you want to do and what the organization does, you can be competitive.
As for your point about the usefulness of a global health or international health MPH, I would say it depends. I still strongly believe that in general, an epi or biostats MPH has the most concrete skills attainment, but it really depends on the scope of work of the organization and the position you will have. During the course of this internship, I know that I definitely want to use my epi skills, and this organization, while they have an emphasis on monitoring and evaluation, they do not have the capacity to carry out their programs in a methodical systematic way that allows for true epidemiologic analysis, a way in which I have been trained and like to work. As a result, their reporting is limited to if donor indicator targets were met or not and not a full analysis of true effectiveness of their programs. They are, however, working on this.
So, basically it all depends on what you see yourself doing and what the organization does. I think a global health degree can be an asset, if you already have some experience or if your program affords you ample opportunities to engage in international technical work.
Hope this helps a little 🙂