stopped receiving push notifications from the "Messages" app

If any Focus mode is enabled, notifications from certain apps may be delayed or suppressed, depending on your Focus settings. In some cases, you may not receive a notification until you manually open the app. Check whether a Focus mode is active and review its notification settings to ensure the app is allowed to send notifications.

Then please check the following.

1. Check your DND status - Settings > Focus [Do Not Disturb]; Use Focus on your iPhone, iPad, or iPod touch - Apple Support
2. Three primary factors trigger the automatic activation of the Focus modes. Users have the flexibility to deactivate each of these factors based on their individual preferences. 
   1. Automatic Activation: Driving, Fitness, Gaming, Mindfulness
   2. Smart Activation: Work, Reading, DND
   3. Scheduled Activation: DND, Gaming, Mindfulness, Personal, Reading, Sleep, Work, Fitness
   4. Here are comprehensive explanations for each of these three triggers. --> Unveiling the Multi-Faceted Triggers of DND Automatic… - Apple Community
3. Check Bedtime Status - Clock App; Use Bedtime to track your sleep on your iPhone - Apple Support
4. Check your iPhone Sound Volume; Adjust the volume on your iPhone - Apple Support
5. Check Call Audio Routing; Call Audio Routing Bluetooth / Speaker / Automatic
6. Check Auto Answer Calls; Turn off Auto Answer Calls
7. Check the list of Blocked Numbers --> Block phone numbers, contacts, and emails on your iPhone ...

Notification Preview when Screen is Locked, Notification Alerts, Notification OFF, and much more...

1. To get Notifications when the Screen is locked see the pic below which is self-explanatory
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2. To turn ON notifications selectively for apps, go to Settings > Notifications > Scroll Down to the App and turn ON notifications... Change notification settings on iPhone - Apple Support (IN)
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3. Some apps have their own notification settings over and above the iOS notification settings: An example of Instagram Notification settings is given below
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4. Help for Notification Alerts --> No Notification Alerts - Apple Community
5. For VIP email Alerts refer to the pic below
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6. The pic below is self-explanatory
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