I think you're running into either a windows permissions issue, or a browser incompatibility issue. The browser version should request permission to load your local fonts before doing so, but some ad-blocker or anti-virus could be blocking the fonts from loading.  It uses a relatively new feature not available in all browsers yet, but it should be supported in recent versions of Chrome and Edge.

This doesn't explain the desktop version not loading fonts, as it loads them in a different way... I'll keep investigating.