In Bug 1243603, I added os.environment.{invoked,launched}_to_handle telemetry that tracks when Firefox is running and asked (respectively, started) to open various registered file types and protocols. I made this Windows only to scope everything down, but there's no real obstruction to doing this for non-Windows platforms. The Windows-specific piece here is the -osint parameter, but I think checking the command line to see if it's a launch is equivalent.

Technically, this would look like simplifying the -osint handling around https://searchfox.org/mozilla-central/rev/4646b826a25d3825cf209db890862b45fa09ffc3/browser/components/BrowserContentHandler.jsm#1011, I think, and doing some testing on other platforms.