Chrome for Android generates and signs .apk file on the fly using WebPack, when the user clicks on "Add to Home screen" (from menu or install banner) option and if the site has a valid manifest.json and service worker.
Extracting and distributing APK: This .apk can be located and exported to desktop using file explorer tools. For some reason, some of default file explorer tools couldn't locate this .apk file. Once exported, this .apk can be used to distribute in controlled environment..like in enterprise devices, where you can enforce the deceive to have Chrome Browser. If this .apk is installed to devices which don't have Chrome browser, user will get a message saying "Chrome" is required to open this app. Once installed, installed PWA apk can be used.
For distributing PWA apps through Play store, google is streamlining the process. A google engineers repose on building APKs (March-2018), when we reached out to them for our enterprise needs.
"Well done extracting the APK and deploying it, it should give a good experience to end users, but I agree it shouldn't be that complicated to deploy web apps on Android.
We are currently working on a streamlined web apps feature, with which you wouldn't need to manipulate or build APKs. This feature will be available on managed devices using Play to deliver apps"
Alternate options: If you think your user base may not have Chrome or don't want to rely on that dependency, wrapping with Cordova kind of hybrid solution is the only way to build your PWA apps for distribution in app stores. With this option, if the "webpack" in the device has the version 40+, user will get PWA benefits. Otherwise, it will still work as a regular hosted web app.
Update on TWA - Trusted web activity is the official way to pack PWA for Android and its available in Chrome 72 and it also supports private/enterprise web apps as the digital assets validation happens in browser now(it use to happen in cloud, making this solution not possible for private web apps).