Now, to remove the battery, gently separate the antenna from the puffy battery below, and then the puffy battery from the PCB. If you reinstall everything without the battery, the video camera will work and the doorbell will light up (blue for powered on, green for RF link and operation).

But without the battery, the inside doorbell won't ring when the button is pressed. The battery provides power to the PCB when the doorbell is pushed.

16-24VAC is only present with the doorbell switch is open - it goes to 0V when the doorbell switch is briefly closed to complete the circuit to ring the doorbell, during which time the 3.7V battery provides continuous voltage to the electronics on the PCB which close an electronic switch. The battery is charged by the 16-24VAC input.

This is primarily a mechanical failure - The battery is probably in early stage of electrical failure as evidenced by the outgassing, this one is putting out correct 3.7V and plenty of current capacity from my measurements. The outgassing pushes the two halves of the doorbell apart, opening the 24V contacts as shown earlier.