All of my 1,000+ visual voicemails came back on the OLD iPhone with the dead sim chip. With your DEAD sim chip inside, turn it on and go through the menu options to set it up as if you purchased it / when you get to the screen that asks you if you want to activate it as a new iPhone or set it up from a backup, select the iCloud or ITunes (wherever you have your backups) and restore from a backup. After you have this, erase everything from the iPhone you were using previously / Restore to factory settings. You can do this even if you have a dead sim chip.
Take your older iPhone and MAKE SURE IT HAS the latest iPhone operating system that your current iPhone has. While your visual voicemails are not stored on your sim chip, they are connected. Your old dead sim was connected to the living room - your newly activated sim is connected to your family room. Maybe AT&T, when they activate a new SIM, puts you into a “new” voicemail box “area.” Think of it like a house.
For some reason, your OLD “DEAD” SIM card is still connected to your Visual Voicemails. Was on the phone with Apple and AT&T for 8 hours + each blaming the other. Was assured by rep, at least 5 times, that no information would be lost. Some of the LTE features were not available on the older SIM card, so called att to activate the sim that came with iPhone X. Here is what happened to me: had an iPhone X that Had a SIM card from OLD iPhone. This is ALL messed up and no one knows their a**es from a hole in the ground.