You can definitely find entire BBC documentaries that have been uploaded to IA; there's an account named "BBC Video" that has nearly 1,000 videos uploaded [0]. However, not all the videos look like BBC content, and there's really no indicator that this account is actually affiliated with BBC. Its undisturbed existence on IA may be a result of the BBC not being particularly worried about the impact of IA's unauthorized distribution of its content.

IA still seems to handle a lot of content moderation scenarios in a manual, old-fashioned sense, e.g. responding to direct requests by email, as opposed to having built out a form-based process. The article says it's unclear who requested the takedown. If the BBC did indeed ask for it on copyright grounds, it's possible that the current error message ("Items may be taken down for various reasons, including by decision of the uploader or due to a violation of our Terms of Use") is the catch-all message for all takedowns.

If that's the case, it's probably time for IA to come up with a copyright-specific takedown message, so that relatively routine requests (e.g. BBC asserting ownership of copyrighted material) aren't conflated with much more nefarious scenarios (e.g. the suspicion that India's government is trying to censor damaging content)

[0] https://archive.org/search?query=creator%3A%22BBC+Video%22