It's not so much a protagonist thing as a controlling idea thing. The whole movie is about one thing: saving the world. So after the inciting incident that's all they're doing. I like what you said here:
...namely a whole bunch of worthless internal conflict for the characters, even though we know of course that they get their shit together in time to save the day.
You're right, it's not a protagonist thing necessarily. It's just that having one would have provided an axis around which character values/motivations can revolve/evolve. (The other way to do this could have been giving a face to the enemy. I think a massive flaw was the facelessness of the Sentinels. Your antagonist needs to have a core against which you rally as a viewer. James Cameron understood this in the
Terminator films. Singer doesn't.)
In
First Class, Professor X is the clear protagonist. He's trying to point out to the rest of the mutants what they
share with humanity, and this brings them to full use of their powers. Eric, on the other hand, is trying to show mutants that they are
better than humans because of what
differentiates them from humanity. The whole movie is about that. It's what makes certain scenes so powerful. The driving force of the wonderful scene in which Eric turns that massive dish is just this conflict. This very same conflict is why we feel like shit when Raven goes Eric's way.
There is no such conflict in
Days of Future Past apart from sheer survival. It's a save-the-day movie from a third of the way on, and even the inciting incident is a save-the-day spectacle, but it works because of the mystery surrounding it.