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.
Just curious, what do you do with the increasing number of companies that use push notifications as a form of advertising venue, and how do you differentiate the security warning notification from your camera app from their special weekly annual sale notification?
The marginal cost of each notification is so low that companies simply spam users nonstop, and their A/B tests shows that the revenue is increasing. What's being lost though is that we're getting more and more agitated with these brands and their uncapped malicious behaviors. This is also true with their UI and UX as well, they keep adding banners with incredibly small close buttons, because someone will continue with the shopping after accidently missing the tiny button, and that's an added sale, who cares about 99% of users who are fuming with dissatisfaction, what are they going to throw away their $200-300 smart home device because companies abuse them?
On my Android, I aggressively mute or uninstall any apps that send push advertisements. I haven't seen one in more than 6 months now. If they think they need to advertise to me, I think I need to do without them.