I use gmail, the biggest problem with the app is the phone puts it to sleep, once it does the Android 10 update it resets all your do not sleep apps (why does it do that?) so it starts putting the app to sleep again, add it back to the do not sleep apps and it works perfectly.
as for the checking for emails notification it is a silent one so it's not really a problem unless you are looking at the phone and in that case a swipe down shows it isn't important, but the checking for mails doesn't flag up on my Galaxy watch or make a sound or even show up any time other than when the screen is actually on so the secondary notification isn't even really and annoyance it's just a minor quirk of the silent notification system letting you know any time an app is doing something.