So, if the recording sits at the carrier level and not on the phone, how does jailbreaking help?
By JBing the phone, it allows you get access to where the AMR file is so you can substitute a custom recording in place so that when you approve the new recording as the final step, the custom recording is uploaded to the carrier's server.
On my JB phone, I was able to locate the greeting.amr file and dropboxed it to my computer and played it back.
I see no reason to think the rest of the steps outlined won't work. I do not believe the directory location is accessible if you have a stock iOS phone however.