Evidence shows that Callin did in fact exit private beta and open to the public shortly after the August 6, 2021 recording.

On September 2, 2021, TechCrunch reported that Callin, David Sacks’ “social podcasting” app, “launches today on iOS” with a $12M Series A, indicating a public App Store launch rather than a closed beta. (techcrunch.com)
That same day, David Sacks published a Medium post titled “Meet Callin, the First Social Podcasting Platform”, stating: “Today is the public launch of Callin … it’s available now in Apple’s App Store.” (medium.com)

Fortune’s coverage also notes that Callin had been in closed beta until this launch, confirming that September 2 marked the end of the private/closed beta period. (fortune.com) Built In SF similarly describes Callin’s private beta and says the team was “excited to launch Callin publicly,” again tied to this launch. (builtinsf.com)

Since the app moved from private beta to a public iOS launch on September 2, 2021—less than a month after the August 6, 2021 episode—the prediction that Callin would open up to the public “soon” after the recording came true.