I agree with much of what is said here, however, there are some details that have been overlooked:
Analog TV broadcasts are transmitting strictly in our atmosphere, as well as being received on antennas as small as rabbit ears on top of a TV placed indoors. NASA was (allegedly) receiving signals using large and much more sensitive antennas (outside).
Also, Australia was receiving and forwarding video when they were facing the moon, not just in Houston.
A side note: out TV signals do go out into space. All radio and broadcast TV signals have a portion of the signal head out into space. E.T. could be watching I Love Lucy reruns right now, if he has a good antenna.
However the lunar module's antenna is definitely too small and short to transmit a video signal very far, if at all.
Ham radio operators can send audio halfway around the world with just 100 Watts of power. If you have a good antenna (on both sides), high power is not required.
Now that being said, it is not enough to counter the point being made in this video. I think the lack of a delay in conversation was also missing. Astronauts in orbit have a time delay longer than what was seen with the (alleged) astronauts on the moon.
And one other thought that just popped in my head: If the astronauts were transmitting using 4Kw power, they were sitting much too close in the module to that antenna. Pretty sure they would have been exposed to a very high level of EMF. Much too high to have been acceptable or safe.