WebView with service worker (what are ServiceWorkerController and ServiceWorkerWebSettings?)
This is only a partial answer, but hope it helps some anyways:
From what I can tell, the ServiceWorkerController is the Java implementation of ServiceWorker for WebView, which is separate from the Javascript implementation. The two do not seem to be tied together in any way.
The Java implementation is a singleton which persists and affects all WebViews in an application, and its only method is to intercept requests via shouldInterceptRequest. It is very similar to WebViewClient, the latter of which has more functionality and only affects the WebViews it is attached to (via setWebViewClient).
The link you posted does seem to indicate that WebViews do support the Javascript version of ServiceWorker. Did you by any chance miss enabling Javascript through WebSettings.setJavascriptEnabled? Also, ensure that you are targeting minimum SDK version 21 as ServiceWorkers are only enabled in Android 5.0 and higher. I am interested to know if you or anyone has managed to get a Javascript ServiceWorker functioning inside a WebView as well.