Admin is right: it's an AT&T (read: AT&T-service-related) issue as the other features are working. But there's a small chance you might be able to finagle a fix on your own.
I assume the AT&T support provided you with the correct APNs to use. But if they were of no help as I understood your post to express, perhaps they didn't. T-Mobile has always been quick to do this and even post theirs on their support site, so maybe AT&T do as well.
Unfortunately I haven't time to research this much to help you, but this site looked promising at a glance from a duckduckgo search:
https://forums.att.com/t5/Network-Cover ... -p/3467513If the APN settings are correct, perhaps you can try a different messaging app. There are multiple apps that use the phone's built-in SMS/MMS system via the APNs' API.
Alternatively, it may be that there is some form of energy savings setting that is causing the phone to sync inbound messages on a slow schedule, say maybe every hour or three. I'm not sure if such a setting exists in these phones as I haven't played with such setting, but perhaps you've found it and didn't realize it got changed? (I'm grasping at straws for this last one.)
Hope this helps.