The easiest way for everyone involved is... using GIFs as GIFs.
The 19% in savings (when converting to lossless WebP) aren't worth it. Not even 60%, when you consider what you lose in usability.
Anyways, the real insurmountable problem is animations with transparent backgrounds. There’s no good way to do this in mp4, and WebP seems to be the only format that does it well. But because of all the little pains, headaches, and surprises that can arise due to subtly different use cases, I still think WebP is the format that will give you the most benefit with the least drawbacks for silent animated images. Even without having transparent backgrounds as a requirement.
If you don't need transparency, I would 100% stick with mp4. Just make sure you set the tag elements "autoplay loop muted".
Mobile Safari used to be pretty weird. If you wanted autoplay (and NO controls), you needed to make sure to remove the audio channel on the video. Setting the "muted" attribute wasn't enough.