melt777,
The library is only intended to work in non-listening modes, which your testing confirms is true. It should only work when the Argon is connecting, or connected to a WiFi network.
As far as it not working in Scenario 3, your brower seems to be disconnecting the device prematurely:
0000413889 [wiring.ble] TRACE: Disconnected by remote device.
That’s why nothing shows up on the webpage. I might recommend trying it on another computer or Android device to see if the behavior is replicable.