First, I agree that films will stand the test of time and provide a powerful and formative impression of what it means to be human. Like other stories that have endured. But I would question whether the stories that are canon became canon because trained or skilled people decided they were canon. I think they became canon because the humans gathered in the dark urged "tell this story" or "tell that story"--and they asked for repetition of what they wanted to hear because those stories spoke to them, touched their hearts, and inspired or motivated or comforted them in some way. Scholars can do no more than recognize canon; they don't dictate what it is.
That being the case, it's just too soon to have a canon of films, perhaps, but the best step would be for the adults in the community to share the films that they love with the next generation, so they have a chance to love them, too.
As far as classical pedagogy goes, I'm strongly in favor teaching in the poetic mode for most of the years allotted to K-12. Exposure--a single viewing of a film--is no basis for analysis. I'm old enough to remember when our viewing was limited to what the networks were showing, and we looked forward from year to year to that one opportunity to see a favorite film or show. It's one reason I think that beginning analysis with fairy tales is so wise--kids are fully familiar with Beauty and the Beast or Snow White, so analyzing doesn't interfere with their poetic appreciation.
Assuming that teaching in poetic mode is a classical objective, I can imagine creating a film culture in a school by mimicking my childhood limitations. Showing one film per quarter, it might be a school custom to view the same films each year (I'm just thinking of the upper school here--say, 7-12--and think how carefully you'd have to select your films). The first few years, you'd just enjoy them perhaps. And then maybe in 9th grade, you'd be asked to write about your favorite character--which one would you want to be like and why? In 10th grade, you might be asked to create alternative dialogue for a scene or write a scene that occurs off-camera. And then only in 11th or 12 grade, when you've seen the films enough times to really know them, would an analysis be attempted.
I'm not sure the assumption that literature or films are meant to be analyzed is consistent with classical objectives. I've pretty much absorbed Norms and Nobility as the seminal work on classical education in our time, so my assumptions tend to come from there.
I tend to think we should apply the same principle to films--to show them to our students and let them have their effect. There is a role for a teacher here, but films--and literature--were made to be enjoyed, not studied. In the poetic tradition, we have to let that level of knowledge happen first.
And I've taken the discussion in a different direction from the original question--sorry about that. Unless "what other questions do we need to ask?" covers it. Because if you were going to do what I suggest--choose a small collection of films to become a part of your school's tradition, what would you choose?