No, you cannot make them independent. You can't have a button and an LED act independently with only one wire.
If you've only got two wires, total, pressing the button will do something to the LED, no matter how you wire it.
I don't view video, so I can't comment on what they did in the video. Could they have 2 wires plus ground? That would let them use two pins like people normally do.
The I/O pin alternates between being used as an input (sensing button presses) and an output (lighting LED). It's not used as both an input and an output at the same time.
The video shows the LED is used to indicate when button presses are accepted or not. The LED flashes always occur after the button presses are finished. There's likely some minimum allowed time between button presses. Once this time has elapsed the I/O pin switches to output (LED) mode.