https://loficities.com/tokyo/?obs&weather=leaves
I was also able to omit ?obs and use the website interface controls to configure it via Plash browsing mode and then the interface went away after a moment of inactivity.Of course, ymmv. But give it a try.
There's already subtle parallax in iOS etc.
Animated wallpapers could be subtle, like raindrops, or animate only when interacted with, when you're bored or waiting for the AI to finish a run etc. :)
Either way, really nicely made, I love it! Small change request: adjust the CSS to make the controls look less vibecode-y.