I had migrated my desktop system to a new iMac about a week ago. Most things worked well, but one frustration was that text messages were not appearing promptly. Messages sent via iMessage appeared simultaneously on the new iMac and my other devices simultaneously, but ordinary SMS text messages were not showing up. I’ve seen this problem before, but I had forgotten the solution.
Anyway, I did find the solution this morning. When I checked the Settings->Messages entry on my iPhone, I saw a ‘Text Message Forwarding’ entry. Tapping it I saw that it was turned on for my iPad and MacBookPro but was off for the iMac. So I flipped it and tested it by logging onto a site that still uses text messages for 2-factor authentication. Sure enough, the message showed up on the iMac. Note that this setting is under the Messages app settings, not the iClooud->Messages setting.
I bought my wife a new Mac, and it had the same issue. I spent hours trying to play with various settings, looking at the Console app, checking Keychain, etc.
The problem was with her iPhone. I needed to let the phone know that this new Mac can send and receive SMS text messages. Went to her iPhone, Messages->Settings, and there it was. I wish Apple popped up a message box on the Mac when this is the issue. Would have saved me hours of googling.
Other thing I had to do was sync her messages onto her new Mac. I had to go to iCloud on her phone and made sure iCloud was backing up her messages (off by default), then hit the sync button on her Mac under Messages->Preferences.
Lots of this is automatically done when you get a new phone, but not a new Mac.