I have had a similar issue where I received notifications and badge count but had to pull down to refresh - whether on the individual account inbox or tue unified inbox. The emails would always appear when I pull down to refresh.
this is no longer occurring for me on 13.3.1 b1. Notifications display on the lock screen or Notification Center. When I tap on a notification it takes me directly to (and opens) the particular email. Fixed. Also, if I open the mail app by tapping on the Mail icon, any new mail is there waiting for me, regardless if I am in the unified inboxor an individual inbox.
Every now and then, when I tap on a mail in the inbox list to view it, it will open correctly, but the temporary drop down notification appears just after the individual mail is opened. Weird. Finally, each account now refreshes consistently when I open the mail app. Up to and including 13.3 release version, I would often have to force close the app then reopen to make it actually fetch mail - for my iCloud account. )The one Exchange account has always behaved correctly....)
So, very nearly there for me. Usable again at last. Only three months after initial release. Good job Apple.