lost media with no hope of being found?

One piece of lost media I cant really see being released (at least in full) is The Thomas and Friends Institute of Innovation, a short film made by Studio Hansa that was for investors and shareholders who want to partner with the thomas brand, only small clips from the short have been made public, the reason for why I dont really see this being ever released in full would be for a quote from the art director of the film, saying that "Sadly due to the confidentially of the project, everything and what's on the site is as much as we can share about the project, by all means."

Might be stretching the definition of "media" a bit, but one thing that immediately comes to mind is YouTube annotations. If you think about it, they kind of were an integral part of many videos for a large part of the early years of YouTube.

One particularly notable example is Jeeper's Media, which was sort of infamous for having a million annotations on every video. I thankfully actually had somebody on the Discord server archive them for me when YouTube announced they were going to kill annotations, but that still means there are countless thousands of videos on YouTube with the annotations completely wiped. Unless somebody has access to some server somewhere at Google, what is lost now will probably be lost forever.

Probably London After Midnight. Even if any copies do survive, they’re likely extremely damaged beyond restoration or salvation of any kind, way, shape or form.

I’d also say the Original cut of Event Horizon. The closest we’d get is probably unfinished workprint footage, even if the scenes were farther in development before being cut. The fact it was pretty recent (1997, less than 30 years ago), makes it all the more eerie and disturbing.

Backyardigans pilot

Don’t they usually have actual cameras at these kinds of events to record them? If so, I wonder what happened to the footage those cameras captured.

Almost all British TV before 1947 or so. Most TV from that pre 47 period in general. (There are some exceptions, most notably from German TV, but they're exceptions)