You are correct. All iPhones from the beginning of iPhone time 17 years ago disconnect from Wi-Fi 30 seconds after the phone screen locks, to preserve battery energy, unless the phone is connected to power. This is done because a Wi-Fi connection uses power continuously, whether data are transferring or not. Cellular connections use more energy, but only when actually sending data. There are a few exceptions:
- If you are streaming audio or video Wi-Fi will stay on
- If you receive a notification Wi-Fi will reconnect for a few seconds to process the notification
- And, I think (but I’m not sure) if an app uses Background App Refresh it can wake and use Wi-Fi; however very few apps actually use Background App Refresh.
However, that has nothing to do with your not getting notifications, because notifications use the cellular network connection of the phone, not Wi-Fi.