mp4's can be embedded just as easily as gifs, are compatible with everything under the sun except for obscure RMS-approved FOSS distributions, and are 'understood' as well as anything (whatever that means.)
I can't think of a single rational reason to use a gif rather than an mp4 for something like this. The reason it's done is because some people are old dogs who can't learn new tricks. There is a good reason sites like reddit will transcode user-uploaded gifs to either mp4s or webms (the later of which isn't supported by Safari, but mp4 has no such problem.)
If your webmaster wants to use a 32MB gif because he can't be bothered to figure out how to embed an mp4, fire him on the spot. Serving up 32MB of gif to every user who visits the page because you can't figure out the <video> tag is inexcusable. This is very far from rocket science.
Sigh all you want, just don't use gifs. It's an obsolete format with no remaining niche. Your first argument for the use of gifs focused on the experience of users seeing an animation "they wanted to show an animation and most people can load it." If the experience of the users is your concern (and it should be), you should feel obliged to never use a gif for something like this because the users will have an objectively superior experience with mp4's in every respect.
If instead you're screwing the users just to save the webmaster some minor hassle, you should sack the webmaster.
I can't think of a single rational reason to use a gif rather than an mp4 for something like this. The reason it's done is because some people are old dogs who can't learn new tricks. There is a good reason sites like reddit will transcode user-uploaded gifs to either mp4s or webms (the later of which isn't supported by Safari, but mp4 has no such problem.)