Then why did they put them on the public Internet in the first place if they were so worried about people getting access to them? If you don't want information getting out to the world, don't put it out there. Major no-brainer. Basic information theory: if you can see it you can copy and save it. To deny it is to deny reality. The AAPB's so-called "few 'irresponsible' Sesame Street fans" are the AAPB themselves for not understanding this inconvenient truth, not their users.
And if they didn't want people downloading them they shouldn't have even thought of submitting them to Internet Archive or Youtube. The thing is, if you view something on Internet Archive or Youtube then it's being downloaded to your disk whether one realises/understands it or not. It doesn't take a computer scientist or an evil, sinister "hacker" to open an HTML page in a text editor and search it for an MP4 (or what have you) link*, then do "save link as" -- which is all being done automatically in the background by your browser anyways. Archive's various web players/emulators function by buffering media files to your terminal's disk or RAM and chase-playing them from your local buffer. It doesn't actually play the file directly from their server. Youtube's player works the same way. Even on "stream only" items on Archive there's usually an M3U playlist file with soft links directly to the media files, hidden in plain sight. That's not a "security" oversight - that was by design, to facilitate its use with external players like VLC. Pop that file open in your favourite text editor and you've a list of downloadable URLs to all your undownloadable files. I mean, duh. Youtube download tools; well, gee, those have existed for years, some work better than others.
[ * It's actually much more complicated with Youtube but there are many, many sites that access in exactly that manner. ]
And the "plugin": probably a widely-used command shell programme called youtube-dl which, despite its name, supports hundreds of sites. It used to work great but the author has gone MIA and it's fallen into disrepair. Youtube-dl is one of the tools I used in at least some of my aircheck preservation work (because we all know how Youtube are these days, don't we?). The fork yt-dlp picks up where it left off. I haven't used dlp (yet, my version of Python is too outdated and awaiting upgrade) but apparently it fixes a lot of problems "classic" youtube-dl has developed vs. updated site infrastructures and adds many features.
But one doesn't even need to do that: if one is dumb enough to still have disk caching enabled in their browser in 2021, then one only need go to about:cache and it'll be listed in there along with the contents of every other Web page you've accessed since the beginning of time. IIRC the file is called "videoplayer" or "videoplayback", but I may be wrong since it's been many years (meaning: well over a decade) since I've had to do it that way.