I think a lot of this could be solved by just better optimizing gifs. A lot of gifs are created by naive tools. Lower the number of frames per second, make better use of the transparency letting through previous frames. You can usually get a gif down to a couple hundred kb.

What gives gifs their staying power is how easy they are to save and repost even on mobile. Videos lose this ability and thus their virality.

The reason gifs live on is because they are so easy to share. Beyond that, the limits of the format are kind of the charm of the entire thing.