The AOSP does not run modern Android apps, since Google has been replacing the APIs those apps require with closed source versions (that only ship as part of Google Play) for over a decade.
That is not true. AOSP can run all modern android apps, and google play APIs are provided as alternatives, but not replacements, for AOSP functionality.
No, that's a widespread misconception. Google Play services is nearly entirely APIs for using Google services. It hasn't replaced AOSP functionality and the AOSP APIs are still there. If you want to avoid Google services then there isn't much that's useful in Play services. Getting developers to not depend as much on Google services such as supporting UnifiedPush in addition to Firebase Cloud Messaging leads to those apps no longer requiring Play services for that functionality.