You're conflating "push notifications" with "being alerted about push notifications." I have many "important but not urgent" apps on my phone configured to just silently add their push notifications into iOS's notification center.

With an app configured to do notifications like this, no banner shows up at the time the app's notifications are delivered; and these notifications don't even show up visibly on the lock screen. You only see this type of notification if you choose to actively scroll down past the "timely" notifications that do get delivered onto your lock screen, to "catch up" on all your notifications.

Basically, these notifications are relegated to an "email inbox" that you can check or not check as you like. But unlike your email inbox, you can go "inbox zero" on your notification "inbox" whenever you like without worry, since notifications (unlike email) are inherently prohibited from being a critical path in an app workflow.