The easiest way to do this is this...
Install a PWA plugin. Here is a free one... (No Pro Needed)
Now, if you want to make it onto the Stores, you will want to write a wrapper (or buy one). There are many options for this, but they tend to be custom.
A cordova wrapper is one of the best options for native wrappers with high quality results. They are generally customized extensively. There are cheaper, pre fab type options available as well. (Some rumors of bans for cheap wrappers continue to exist, most in the field have reported these have stopped by the worse offender, Apple who has slowly (and now completely) joined the PWA bandwaggon).
I recommend PWA's as a consultant. They make sense. PWA's offer ALL the features of traditional Apps with the same code that runs your site. A wrapper + PWA offers even more and can provide very good control over the offline content loading. Push Notifications are available for both. The rest is just Gravy added to the feature list.
STORE ACCESS: I believe you may put a PWA app on the Microsoft store. The Android/Apple stores tend to require a wrapper. A wrapper is essentially a browser in a window with an Icon.
For a long time, Apple would kick people from the store if the wrapper was not substantial improvement over the web experience. They now tend to be much more lenient as long as it works well.
There are a few companies that make pre-existing themes that go into wrappers. You can also try these if you want the store. This is a much better, and cheaper, way to approach app development and one I specialize highly in.
The rest API as specified in the question can also be integrated but should not be needed for this approach.
Push Notifications are available and requires another free app and custom code to integrate with the system you build. There maybe a service fee for insane amounts of push notifications. A lot of Free options are also available, but integration tends to cost.
My OVERWHELMING conclusion:
Overall, the PWA+Wrapper approach offers everything if you want the store presence. I love using the same logic that I use to build the site in the app. No need to duplicate efforts. Then I customize the site experience to be mobile friendly. Finally We build a wrapper to consume the website. We can offer offline access, notifications and store access cheaply with this approach. It offers you a single code base and user experience that is seem less with reduced development costs of a dedicated app for multiple systems.
SuperPWA's degrade nicely. Apple, Google, Firefox and Microsoft all fully support PWA's now. Any report otherwise is out of date.