Hello this happened to me i was devistated to have a 20 year old greeting lost out of the blue.
I found it with some hints from tmobile support. They didn’t help me, said they couldn’t recover my greeting but while trouble shooting they showed me a method to enter your voicemail inbox. Using the phone with the missing greeting: Either call yourself then press *(star/asterisk) OR just hold down 1 on the dialpad to call ur your voicemail. Then ur password to your inbox should be the last for digits of your phone number. Then after you listen to all ur pending voicemails you’ll arrive at the main menu. The first option should be “to listen to you current greeting press 1. “ In my case by the grace of the lord almighty, it played my original voicemail. I kept pressing replay until i could grab another phone to record my greeting on the voicememo app. I now can use this recording by creating a new custom greeting in iphone. That’s how I did it, i hope this helps anyone else that shead many tears to their lost greeting.