I'm only going to say this one more time. He is indeed a governor as well as a military figure.
If what you want to say is you don't care that he sees nords as barbarians, which could affect his relations with them, especially if found out, then that's fine. Perfectly fine. I think it is something to consider, considering someone that thinks that way isn't likely the best person to put in charge of a war where part of your job requires gaining Nords to your side. All he'd be able to say to them is it's their duty to serve the Empire, and he can't speak to them on a level they can relate to.
You say it doesn't matter because he doesn't speak to them, but the fact that he doesn't, whether by choice or by limitations IS a flaw against his capacity as a leader, since his side is almost faceless, whereas Ulfric is well known and charismatic. That's an advantage Ulfric has over Tullius.
You cannot pretend it's not a point against him by ignoring that PR is a part of his job. Not doing your job doesn't mean you're not bad at it.
Babysitting Elisif, a Jarl, is exactly that. As is his treaty negotiations.