Push notifications from the mail server are completely independent from the downloading of the email message.
Don’t believe me? Try it when on a flight that offers free WiFi messaging (e.g., United). You will still get push mail notifications, but the email app will not be able to connect to the server to download them due to service restrictions. Pay for full WiFi, and the messages are downloaded as expected.
I think most of the people having email issues are actually having connectivity issues, due to either slow or unstable data service, or issues caused by private relay or a VPN. Some say resetting network settings can help fix it. For others, it requires removing a WiFi configuration and setting it up again.