I uploaded/installed the latest Divi theme zip which was a fresh download, and got a broken theme error saying the stylesheet was missing after activating the theme. I checked the theme files and none of the required files are missing, including the stylesheet in question. Everything was present and correct.
To compare, I uploaded/installed the same Divi theme zip into another local site using DesktopServer, and the theme works perfectly, which makes me think it’s a problem or bug with Local.
Can you try opening your browser developer console to see if there are any specific errors on those assets? You should see if they’re 404’s, 403’s, 500’s, etc.