Just to update this thread regarding the gamepad input issue - Valve got back to me about this, and it turns out that Spacewar - the title with app ID 480 often used for testing purposes - has a custom Steam Input configuration which appears to be why gamepad input doesn't work. I think that means Steam is taking over gamepad input and expecting the app to use Steam Input APIs, but in a Windows WebView2 export it's just using the regular browser Gamepad API, which inside Chromium is using the regular Windows gamepad APIs. So basically it's a configuration issue.
If you have your own app ID, you should be able to configure Steam Input for that app in the Steamworks app admin - I think you just want to turn it off (from the documentation it looks like you can choose 'None selected') or use a default mapping. As we don't publish titles ourselves we don't have an easy way to test this right now - perhaps someone can give it a go? Hopefully this means you can both initialize the Steamworks SDK and still use gamepad input (providing you set "allow-host-input-processing": false for now until the WebView2 bug is fixed).
I've also asked Valve for advice on how we might be able to set up a testing mode where you don't need your own app ID to check Steamworks integration and gamepad input are working - hopefully we can make this a bit simpler to try out.