We’re in agreement on the physics of rocket propulsion. However, “fire” is essentially defined as a chemical oxidation reaction. The reaction itself doesn’t have mass. While fuel and oxidizer undergo the oxidation reaction, it isn’t the reaction itself providing the propulsion, its the mass and velocity of the combustion products.

This is why the “natural element” definition is old and out-of-date. Any discussion of “fire” as an element is a philosophical or literary exercise, not a scientific one.