That's how I would explain how rockets fly using Newton's third law:

The action:

The rocket throws tons of fuel in the opposite direction of flight.

The reaction:

The rocket accelerates in the direction of flight and gets faster.

→ Nothing else is needed to explain why rockets fly!

.

Now tell me:

Is there a flaw in my explanation?
(If yes give me a simple explanation of it, please.
The explanations in the video seem counter-intuitive to me.)
A flaw in those five lines should be easy to find for you, right?