I believe that API is in the play store itself. If your user used a third party store, you could ask them to give you an API to install other apps with no user interaction too.
As far as the platform is concerned, any app with the right permissions can install any other app. It just so happens that only the play store and other stores have that permission.
That's true; you're right that there is a open platform between the AOSP layer and the APK apps that run on top of it.
As you point out there is a fly in the ointment (which is enhanced by Google making it very difficult to ship with anything but the Play Store).
As an app developer like Spotify, the platform is the base of devices out there in the wild and how I build on top of them which, for me, is inclusive of the play services and play store.
Android definitely is a lot more open than iOS. But when the Play Store enforces a mechanism that favors Google, it's not of a practical difference to my livelihood than when the App Store has policies that favor Apple. If you have a really loyal following, like Epic/Fortnite, it's better than nothing. But even then the barrier to your users is huge and even the most establish companies with technical user bases will have a hard time crossing it. Nevermind the fledgling startups or developers with non-technical user bases.
As far as the platform is concerned, any app with the right permissions can install any other app. It just so happens that only the play store and other stores have that permission.