Nope they're completely independent of each other. We've had push notifications on iPhone well before Background App Refresh was a thing. I've actually talked to many people who thought the same thing as you though. Perhaps Apple needs to explain it a bit better to consumers.Doesn't push notification require background app refresh to work? If the app is unable to send and receive data while in the background, how would it know what to push?
Background App Refresh "pulls" data constantly in the background (a somewhat intensive process compared to a push notification). "Push" notifications do not constantly pull data checking for an alert. The alert is "pushed" to the device by the server and then displays the notification. Hence the name.
So yeah, feel free to turn off Background App Refresh. It will not affect your push notifications from other apps at all.