Biden's "hot mic" moment was an intentional slip. It is something relatively easy to plan, especially when the questions from right wing media are so predictable. This moment was done to add a message, something like "Biden is a real person, aside from his station... and is frustrated by the media". It is also done to add drama and fire up his support.
Yes, Peter Doocy is a stupid son of a bitch.
Yes, the whole thing was intentional.
You can absolutely enjoy the moment. Just don't be taken in by it.