So, yes, the blockchain doesn’t make files smaller, but it could work to verify their authenticity, and that they have not been tampered with.

As with every other suggested use of blockchain, there are already better ways to verify contents. It's called hashing, it's been around for decades, and we do it all the time.

So instead of videos being hosted on 1 server, videos could be downloaded and made available by anyone to anyone at any time.

This is going to run into all kinds of bottlenecks. Individual users may have a fast enough Internet connection to stream HD video, but uploading is often much slower. Even if not, one user could only co-host maybe 1-2 other users. Also, ISPs sure aren't going to like all the increased bandwidth!

People always vastly underestimate the bandwidth requirements for smooth, streaming video.