Same problem, nothing works out! Such a nonsense.
Looks like for my case if I received 3 mails when I open the gmail after opening it lists the emails. Then plays the notification sound.
I tried testing by sending email to my gmail from a different email id too. Then after 10 minutes disconnect the recipient device from internet and open gmail. The email doesn’t show.
So the issue is it is not bringing the email upto the device from Google server in BACKGROUND.
All background settings and everything else discussed here is double-checked to be correctly place. Tired of the issue!!!