You could try going into Safari Preferences, Privacy, and then "Manage Website Data..." and delete any instagram cookies you see. It could be that a cookie is corrupt and is causing the page to load incorrectly. It has happened to me on other sites, so it would be worth a try. You will need to sign in again to instagram.
From what I can tell from searching, Instagram Stories doesn't appear to work outside of the app. There is an extension you can try called "Chrome IG Stories," but it's only for Google Chrome. I don't know how well it works, but it sounds like it's been this way since 2016, so I'm guessing proper browser support may never arrive.
I see the same behavior. It always starts playing if I click on the blurred "pre-loading" image and very occasionally it will autoplay. I made sure that I wasn't auto blocking video on the instagram site.
EDIT: Autoplay of stories works just fine for me in Chrome, but also fails with Safari Technology Preview.