Converting a WordPress site into a PWA

I have a WordPress website and I am transforming this website to a PWA with the help of a plugin.

The PWA is working as expected: notifications, working without internet …

I have also followed this tutorial completely:

https://css-tricks.com/how-to-get-a-progressive-web-app-into-the-google-play-store/

At the end, I have an Android launcher.

The new app is doing this:

1. If the browser is not set by default, it will ask the user to choose a browser.
2. I have an URL bar at the top of the app.
3. The app is not working without internet.

How can I fix all those points?

1. If the browser is not set by default, it will ask the user to choose a browser

Can't get this one, every mobile phone already do that by default, I think.

2. I have an URL bar on top of the app

Check if you have included the tag:

"display": "standalone"

on the manifest.json file

3. the app is not working without internet.

Check your service-worker.js file to see the resources needed to provide an offline experience are effectively cached. One can use this nice tool from Google to define routes policy easily.