So I looked into what's involved in hosting the videos myself. It turns out the hit from hosting the videos myself wouldn't be too bad. Since the videos were video slideshows, it turns out they actually compress really well. I played with different ffmpeg, and I found that I could drop from the 50MB files that my video program produced to 5MB files by using two passes, variable bit rate, and a large maximum duration between key frames.
Now the second problem. There are two main video formats on the web: webm and mp4. Apple owns patents on mp4, and they purposely refuse to support any video formats except mp4 on their devices so that anyone who wants to provide video content to Apple users must pay for Apple's patents. I couldn't just use ffmpeg to transcode my videos to mp4 format because it doesn't come with a proper patent license (licenses are needed to decode and encode the h.264 video and AAC audio). I tried scouring around the Internet for a properly licensed version of ffmpeg that I could buy, but I had no luck in this. I could have just purchased a whole new video program with its codec packs, but it's hard to tell whether the codecs that come with a video program would expose the tuning parameters I needed to get the small sizes I wanted.
In the end, I went with a cloud transcoder since they presumably purchase a patent license for their services. It turns out most of the cloud transcoding services have gone bankrupt, so there's only a few big ones left like Amazon Elastic Transcoder, Zencoder, and Telestream cloud. Initially, I was leaning towards Zencoder because they were pretty upfront about the fact that they let you set all the ffmpeg parameters yourself and they said they 2-pass encoding. But the system seemed sort of messy--you needed to copy your files into s3 and give them read rights to it. At that point, it seemed easier just to go with Amazon since I already had an account with them. At first, I couldn't get the Amazon stuff to start, but apparently, the Amazon Transcoding console won't start until you upload a video file to s3 first, which is a little bizarre, but whatever. In the end, Amazon actually exposed the parameters I needed to get my slideshow to compress well, and the final file sizes seemed to be competitive with what I was getting from 2-pass encoding with ffmpeg myself, so I suspect that Amazon must be enabling that but not saying it in their documentation. The web interface requires you to manually enter in the settings for every single file you want to transcode, which is a pain, but I only had 15 videos or so, so it wasn't too bad. It's possible to script the transcoding using their APIs, but I was too lazy to do that.
Actually serving mp4 videos on a website also requires a patent license (separate from the patent license for encoders and decoders). Fortunately, I was offering free educational Internet videos, and those are exempt from royalties.
It's sort of annoying that the technical aspects of putting a video up on my website only took me an hour or two to figure out, but the process of trying to figure out how to do so legally ended up taking two days. I really hate how Apple is doing everything possible to sabotage the web and extract maximum profit from it--they patent important parts of the HTML specification, they refuse to support formats that can be used without patents, they refuse to support new standards in their browsers if it makes them competitive with apps--it's just ridiculous sometimes.