I was skeptical that the takedown had anything to do with Viacom, and possibly that it was something the uploader chose to do after all the publicity it got, but I just saw the takedown email in the Lost Media Wiki Discord server which does name Paramount as being the reason.
It's also interesting to note that they even went as far as archive.org since most companies seem to not bother with the site. They certainly didn't do anything about the dozens of Blue Prints reuploads that were there after the original YouTube video was blocked.
I don't think Paramount manually took it down from Archive. It is WAY easier to mass remove a video from YouTube and there's still many, many, MANY uploads containing the clip. Just because one random Internet Archive upload was taken down does not prove anything. It was probably IA's shitty copyright bot going off randomly at the site of SpongeBob.