I can see this most likely being implemented as the use of VR and, more likely, AR becomes more widespread.
Why?
Virtual (and Augmented) Reality is all about immersion.
With this in mind, music companies are probably already considering ways to make music more immersive so they can capture the VR music market.
Their first instinct is probably going to be 360 music videos (or, even cooler, music videos that you can walk around in and explore).
However, it's going to be really expensive to make 360 music videos for every single song.
So how can you get people interested in the rest of the songs your label is selling?
Album-only bonus content
Right now, we get this in the form of bonus songs.
In the future, maybe this comes in the form of an exclusive animated or interactive album cover.
Back to your question.
Why aren't artists doing this?
- The desire for the content isn't there yet (nor is the funding). Since music is considered to be a "background media" at present, this kind of awesome design isn't needed. In the past, music was more of a "foreground media". For example, think of a family gathered around the radio to listen to a show, or a group of friends hanging out around a record player. In the future, if VR and AR pick up, something visual to go along with music will almost certainly be desired. For more info on music as a background media, check out this article and this one.
Is it a technical limitation?
- Definitely not. There are plenty of artists that have the capability to use a cinemagraph for an album cover. It's more of a "user needs" limitation.
Just a hold out from what's always been done?
- At this point, yes it is a hold out. Media takes quite a while to shift styles, especially if a new style comes with an additional cost. For example, I've spoken with several designers at companies who refused to change their website theme to a responsive one until recently. The only way the designers could finally convince them to change was by pointing out that "everyone is doing it".
- Whenever new designs, ideas, styles, etc. come out, they follow an "adoption trend". Right now if we're seeing any animated album covers, they're being produced by Innovators (usually about 2.5% of the market). Take a look at the graph below. I think it's a fantastic way to visualize how these new trends often come to market, and why they take so long to do so.

Not completely related, but still interesting article on "old design trends".