I have both my work exchange account and gmail account using the native mail app. It was solid in 12.4. They finally fixed the push notifications but when I often get a notification and open the mail app it has to download the email. Before anytime I got a notification and opened the mail app the emails where there. Also sometimes they will show up in the separate exchange and gmail inbox while not having to download into the all inbox folder until I wait a bit. Happened on both my Max and iPhone 11 pro max. Im on the final 13.1