Incorrect. At least according to the Matrix (chat) app FAQs I have read recently.

With Matrix apps, certain metadata is pushed from the chat server, to a push server, through Google and then to my device. But the message is not part of that data - it's E2EE. What happens is the app wakes up from the metadata notification, and then fetches the message and displays it in the notification field.

Your last point is correct, at least until/unless this is remedied in Android, too.