How to download old (pre-iPhone) voicemails?

How do I download voicemails that were saved before I got my iPhone? There are two very special voicemails (birthday greetings from my grandpa, who has since passed away) saved to my voicemail box from back in my flip phone days. I can play them as "saved messages" if I call my voicemail from my iPhone XR, but they don't show up in the visual list of voicemails on the iPhone screen the way new messages do. Is there a way I can still download or save his voice from these older messages?

My carrier said they can't help and I should ask Apple. Multiple attempts at just screen recording while playing the voicemails on speakerphone generated silent recordings. (Yes, I turned the mic on for some, off for the others, same result both ways.) If all else fails, I'll try playing the messages through the computer or a bluetooth speaker and recording the sound with a microphone, but I'd much rather have the original clarity of the actual file. I know it must be a little digital audio file that exists somewhere because it plays when I dial into my mailbox, so how do I find and download that file?

Any help you all can offer would be greatly appreciated!