In other words, why rockets do accelerate in empty space?
Because of Newton’s Third Law? Yes, but this is not a very understandable way of putting it.
Instead, imagine a hollow cube made of 6 square steel plates, one square meter each, a few millimeters tick, well welded together, full of nitrogen gas heated to 300 K, so that the internal pressure is 1 bar. The cube is floating freely somewhere in interstellar space.
The pressure exerts the force of 100,000 N on the front plate, and 100,000 N on the rear plate. The forces are in equilibrium. We will dismiss the other four plates since the same goes for them.
Now we make a 1 square centimeter hole in the rear plate, so that the gas induced force is acting upon it with the force of only 99,990 N. Therefore there is no equilibrium anymore, but the resulting force of 10 N starts to push the cube, until all the gas leaks out.
This is how a rocket works. This is why Goddard was right.
Why the New York Times was wrong? That is another story, they had no clue, regardless of details.