It is awfully true for rockets, not so much for planes.
After a quick calculation, it appears that only about 1% of the fuel is converted into kinetic and potential energy. Which meant that 99% of it is used to overcome drag. Drag is a function of speed, surface area, air density, and the shape of the plane, not its mass.
Rockets are another matter entirely. That's because fuel is used as a reaction mass. As a result, they are constrained by momentum, which is why we need ridiculously large rockets to launch ridiculously small stuff in space. Planes just use surrounding air as a reaction mass.
Weight matters for aerodynamics, and it is a very complex subject that goes well beyond simple energy considerations. A good example is some gliders which have water ballasts to increase their weight. It allows them to glide more efficiently at high speeds at the expense of losing altitude faster at slow speeds. These ballasts can be jettisoned if the flight conditions change, and also before landing in order to limit stress on the airframe.