True, WE need air, but not for the reasons people think. The vid below demonstrates this.
What we perceive as sound is a function of 2 factors; pressure and frequency. Why? Because our ears are evolved so. No movement (ie displacement) of air is needed.
To produce a pressure from the trumpet, we need to apply a pressure into the trumpet/tube. For most of us, this means we must blow air into the tube. But, the tube must resist the air we apply: else no pressure will result.
To produce a frequency, we need to apply a vibration to the tube: for us, the source is our lips. For most of us, we need to exhale air against closed lips to produce a vibration/frequency.
Unfortunately, the science we learn in high school is erroneous, and thus people carry this throughout their lives. Sound is always depicted via 2-dimensional graph as a line to form a 'wave'; this pictorially presents sound to have a frequency we can measure, but excludes entirely the factor of pressure. But here is a simple fact: sound is 3-dimensional and 2-factored.
What we hear as high pitch from a trumpet is a function of both frequency and pressure: ie greater vibration and greater pressure, from the player. To achieve this, many factors must be combined. One factor is the resistance of the trumpet to help us create greater pressure. Thus, a trumpet requires bends in the pipes; the bends assist to create pressure. If the right pressures are achieved, what we call 'partials' can result. This is why a trumpet having a D-shaped lead pipe will be easier to play for high notes than one having a semi-circular C lead pipe.
The prof in the vid below is a talented musician, not a scientist. Unfortunately, he cannot explain the science for what he demonstrates. But, he is on the right track.
For anyone who may doubt the above, kindly ponder why the flame produces a sound exiting the glass tube; when the flame itself consumes air and burns without a vibration.