One of the shows talked about how EC originally deleted the voicemail, then got her carrier to restore it once she found out Faith had been murdered, and then gave it to the CHPD. So they have had it almost from the very beginning.
I think the reason we (and these shows) talk about it so much is just because it's a piece of information that we have access to. CWD was the first to report on it, IIRC, and got Arlo West to analyze it, but they didn't get it from the CHPD, they got it from EC. Then, Arlo West produced this stunning transcript of it that made it out to be some sort of case-cracking piece of evidence, and CWD made it a center piece of their story.
When we all listened to it for ourselves, we were extremely doubtful (the exact one part that matches the transcript for me is the "Ow!" part, and that sounds more like a squeal of excitement than a cry of pain). I don't think the CHPD thinks the voicemail is particularly important, either.