In Chrome, the File Handling API shipped in 102 (and is no longer behind any flag or trial). In Edge 102+, despite what it says in the Microsoft article on the subject, the API is completely broken. Following the instructions there to enable the API in edge://flags (which shouldn't be necessary after 102), the demo My Tracks app does not behave as described (after installation).
I have also tested https://pwa.kiwix.org, which can be installed as a PWA, and will handle ZIM files in Chrome, when double-clicked from Explorer. I tested with Edge stable, Edge Beta, Edge Dev and Edge Canary (104.0.1287.0) in Windows 11 Sandbox. In the first three, clicking on a ZIM file (this small test file can be used) will open the installed PWA, but there is no permissions dialogue box on app launch, therefore permission is not granted to handle the file. Behaviour is the same whether or not the edge://flags are set.
In Edge Canary, a permissions dialogue box does open on launch of the app, but it misrecognizes the double click as a request for two files (!). Giving permission to open the files will open the PWA, but then it cannot deal with the permission.
In sum, it's a mess. But the API works perfectly in Chrome stable with the same app.