I'm having the same issue with a header video on a (7.1) site. Your results @otilia are very similar to my own. Here are my observations (for both your site and mine).
Video plays fine on any desktop/browser and Windows/iOS
Video plays OK on Android mobile (seems OK on all browsers?) - half the fish
I think your video may actually be centered - fish is aligned to the right in the video? Video
Video doesn't play at all on any iOS mobile (iPad and iPhone - portrait or landscape - yours or mine)
No fallback image is available on iOS mobile on your site (Safari & Chrome - portrait and landscape) - your header section where the video or fallback image should be is plain white (blank) background. Your header text and button are showing correctly.
I managed to get my fallback image working on iOS mobile (no video). Wasn't working at all (blank background like yours) and I kept fiddling then suddenly the fallback image appeared on iOS mobile?? Not sure why? One thing I noticed - make sure no default image is selected under the "Image" option toggle for the section background.
My fallback image is not so much of a problem for me - but it would be really nice if Squarespace video headers would actually play on iOS mobile - like they do for other web frameworks. I've tested other sites (Wordpress for example) that include video headers and they work just fine on iOS mobile (Safari and Chrome).
I have quite a bit of custom code (javascript & CSS) added but I'm pretty sure that's not the issue? No conflicts/errors reported in Console. I suspect it is just iOS and the method that Squarespace uses to load videos that is the problem? Pity I can't offer a solution at this stage but just letting you know my experience for what it's worth.