I am frustrated and disappointed with two aspects of this issue.
First, the Help Article referenced ([Short Code: 79913] Not Receiving Texts from Social Security Administration (SSA) – Republic Help) does not provide useful information to us experiencing the problem. No date is listed on the article, the article claims that “you should be able to authenticate your account using email” - but fails to advise that in order to add or change authentication to email (from text message) we must take non-intuitive, non-explained actions to falsely tell SSA that “my mobile number has changed,” so yes, postal mail me a new activation code in 5 to 10 business days. Only by reading through this thread did I learn that is the ONLY solution available. No amount of trying different options on the SSA site provides otherwise, despite hints toward more expeditious paths.
Second: what the heck could take more than five weeks to sort through? My wife and I are both stuck with this issue, only now (belatedly, only after searching in Community for “Social” to find this thread) have I requested the USPS mailed reset code. I volunteer my time and Moto G5+ to track down where the text fails. I hope someone is actually trying to figure out how to get SSA two factor authentication via SMS back working! Should I call my congressman, too?