On the other hand, I believe the <video> element could be The Animated GIF Killer. GIF compression is terrible especially for video, and I'm sure with some tweaking of compression options you could make a .mp4 video with higher quality and a fraction of the size of the equivalent .gif. If there were some site like Imgur that let you upload some small video fragment, let you do the appropriate timeline snips, and automatically created a nice small .mp4 and a fallback .gif, that would be truly excellent.
Give it a maximum video length of 10 or 20 seconds and pre-load the site with a bunch of the more popular reaction GIFs like the Picard Facepalm and Orson Wells clapping, and you should be good to go.
EDIT: Looks like https://mediacru.sh/ does at least the GIF->MP4 encoding, although it misses imgur's handy "here's how to use this on your site" examples.