Why is there no current flowing through another parallel wire in this video?
When the transistor is turned on in that manner it can be switched into saturation if the base current is sufficient. The collector-emitter emitter voltage will drop to about 0.2 V. This voltage will be applied to the LED.



Figure 1. Current through various colours of LED as a function of forward voltage. Image source IV curves.

Figure 1 shows that none of the LEDs from infrared to ultra-violet will pass any significant current at 0.2 V. There just isn't enough voltage to get the charge carriers to jump the P-N junction.



Figure 2. A water check-valve analogy. Image source: What is an LED?.

If you look at the check-valve in the figure above, it should be clear that the spring normally keeps the ball in position and prevents back-flow. When “forward-biased” the ball shut-off can be moved against the spring but it will take some initial pressure to move the ball. This results in a pressure drop across the valve: the pressure downstream will be less than the inlet pressure.

In a similar manner the PN junction in an LED causes a voltage drop. For a red LED it is about 1.5 V to 2.0 V. You need to exceed the Vf to get enough current to flow and light the LED.

The links are to articles by me and may help you further.