The way I see it, the question should not be, "Do video game cutscenes belong on Box?", but should actually be, "What is required of video games cutscenes to make them acceptible on Box?"
All of the responses I have read in this thread seem to agree that cutscenes are acceptable, but seem to give differing opinions on what cutscenes would be acceptable and what would not.
In my opinion, cutscenes should be allowed if they provide information that is relevent to the related anime's plot, and provide a complete and understandable story.
For example, the Derge of Cerburus cutscenes. From what i have read (I have yet to finish the game and therefore have not seen all the cutscenes) these cutscenes do provide information that relates to the Advent Children anime, but from what i have seen of these cutscenes the story is sometimes unclear because you skip from one place to another without ever knowing how you got there, without playing the game some things are unclear. So I do not believe that these belong on Box.
Now the Xenosaga cutscenes (once again I have not seen all of) do provide a complete and clear story, thanks to editing on the subber's behalf. From what I have read about the anime, (have not seen it yet) it seems to be a retelling of the same story, just in a different form. Therefore it seems both the requirements are met, thanks to some editing, so I believe these belong on Box.
The DMC4 cutscenes (which have been rejected prior to this post) do not seem to provide information that relates to the anime. It seems more the other way around, the anime provides information that is relevant to the game. For this reason I do not believe that these cutscenes belong on Box. As for the clear and complete storyline there seem to be differing opinions. Those that have played the gave say it meets this requirement, but those that haven't seem to say it doesn't. This would lead me to believe the complete and clear story requirement in not met either.
So basically what I am saying is that there is no yes or no answer to the posed question. The answer is more of a case by case basis and to what belongs and what does not.
Well, that is just my 2 cents on the matter.