Free and Open Source Software

I built a custom app to do it since I couldn't manage to fire the relevant intents from an adb shell without root.

I lifted the code from AAAD

Specifically the InstallAPK method in MainActivity.java

Intent intent;

            if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.N) {
                intent = new Intent(Intent.ACTION_INSTALL_PACKAGE);
                intent.setData(getUri(file));
                intent.setFlags(Intent.FLAG_GRANT_READ_URI_PERMISSION | Intent.FLAG_ACTIVITY_NEW_TASK);
            } else {
                intent = new Intent(Intent.ACTION_VIEW);
                intent.setDataAndTypeAndNormalize(Uri.fromFile(file), "application/vnd.android.package-archive");
                intent.addFlags(Intent.FLAG_ACTIVITY_NEW_TASK);
            }

            intent.putExtra(Intent.EXTRA_NOT_UNKNOWN_SOURCE, true);
            intent.putExtra(Intent.EXTRA_INSTALLER_PACKAGE_NAME, "com.android.vending");
            getApplicationContext().startActivity(intent);
        } 

Basically you construct an Intent ACTION_INSTALL_PACKAGE with data pointing to the APK file and the extras EXTRA_NOT_UNKNOWN_SOURCE=true and EXTRA_INSTALLER_PACKAGE_NAME="com.android.vending" which tells the installer that this APK is not sideloaded and it's the play store asking to install it.

You might still need to enable unknown sources in Android Auto developer settings (separate from phone developer settings).

If I remember, I'll try to pull the code for my app from my PC and post it.

the paid version is still Free and Open source

As I understand it, the paid version is not free and open source. For example, it includes a weather plugin which is non-free.