Also currently experiencing the same issue

I had this exact same issue for a while, you want to make sure the permissions are enabled on the website or the content script won't inject. I still had the issue at random times, but after updating to the beta 2 it was much more consistent.

When you create a Safari web extension in xcode, you're provided with a SafariWebExtensionHandler file. Which has a beginRequest method. When you call browser.runtime.sendNativeMessage from your background script, it calls that function in your swift code. In order to have access to the sendNativeMessage method in your background script you must add the nativeMessaging permission to the permissions array in your manifest.json.

Updated to iOS 15 beta 3, still same issue.

Yes this is in the SafariWebExtensionHandler class. You can print things to the console and view them when attaching the debug process to the Extension App

iOS 15 Beta 8 Xcode 13 beta 5 still same issue