Why can't I receive verifications codes sent as text messages?

Apple 2FA Verification Codes are normally sent to all of your Trusted Devices via the Secure Popup you’ve seen.

Additionally …

Apple 2FA Verification Codes are sent via insecure SMS texts ONLY to your pre-registered Trusted Numbers; ONLY when you explicitly direct it to be sent; and ONLY to a single Trusted Number at a time.

See these two fairly important articles for additional info on this subject:

Two-factor authentication for Apple ID - Apple Support

Get a verification code and sign in with two-factor authentication - Apple Support

Presumably you’re not receiving verification codes from banks and other web sites via SMS Texts … ???

If this is indeed the situation, check in Settings -> Messages to verify that the specific sender(s) are NOT blocked, AND that “Filter Unknown Senders” is “off.”

Beyond that, you’ll need to contact your carrier to resolve the non-delivery of SMSs.

Well, if you’re still not receiving SMSs, that leaves your carrier.

(They might be getting hung-up in the carrier’s Bulk-SPAM-SMS filtering)

Sorry.

Contacting a carrier usually isn’t a “fun” experience. 🫣

Worth a try. Certainly can’t do any harm.

Probably simply “Reset Network Settings” would be sufficient to make any difference.

Note that this action will also erase ALL “remembered” WiFi SSID’s and associated passwords, so be prepared before attempting.

Settings -> General -> Transfer or Reset Phone -> Reset -> Reset Network Settings

Inadvertent dupe post deleted.

Re: “… AT&T has been NO help whatsoever …”

Shocker ! 😲

Have you tried involving the “sender” … you are after all THEIR customer.