I found a video for "Band on the Run," identified by the uploader to YouTube as the original video for the song. Elsewhere (Songfacts) someone also mentioned this as the original video, claiming that it had been withheld for years. I mention it because the video is very, very, very Beatles-heavy. The Songfacts person was arguing this as proof that the song is primarily about the Beatles.
It's annoyingly hard to find. (OK, not that bad. Search "band on the run original video" or try this: Might be the original video ) To me, it looks like something VH1 put together in the '90s. The sound is odd enough that it could be 40-odd years old; it doesn't sound quite like the single, but not that unlike either. It looks like somebody with a Terry Gilliam fixation got loose in an editing studio.
I just don't see it. Paul apparently never said the song was about the Beatles. If we're believing Wikipedia, he said it was about marijuana arrests. He seemed to be making every effort to make a clean break from the Beatles at the time is how I recall it. Also, it would be nearly 10 years before MTV came along; what exactly would they be making a video for at the time of the record's release?
I think people are just assuming that that is the original video because it should be about the Beatles. Obviously, Beatlemania was a lot more insane than Wingsmania ever became. Obviously, one's experiences affect one's songwriting and the Beatle experience probably informed the song more than a little. But why would he lie about it back then? "It's about the Beatles" would have probably been a much more market-friendly answer than "it's about marijuana arrests." Right?
I just wish I could find a more definitive answer as to whether this is actually an original video from the '70s. About the only argument I can offer in that direction (apart from the weird sound) is that I'm pretty sure that VH1 stuck their logo on everything that they made. But I don't remember that with certainty either.
Edit: The Wikipedia entry for the actual song also notes that the Beatle-oriented video was made at the time of release, with no further details except for the director's name. Not that this is definitive, but it seems to be the best I could do. Perhaps it was an EMI/Capitol move that met with Paul's disfavor. Or maybe the video just went clang because as I noted, MTV didn't exist yet. Go figure.