You are right that life does not provide any source of energy, so any results of life activity could also appear when there is no life, including free elemental oxygen.
What life does, is changing by many orders of magnitude the speed of certain chemical reactions, including the reaction by which water is decomposed, releasing free dioxygen.
The concentration of any substance in our environment is normally an equilibrium concentration, which is determined by the speeds of the chemical reactions that produce and that consume that substance.
On a planet without life, the speed by which the stellar light produces free dioxygen is very small in comparison with the speed at which the dioxygen is consumed by oxidizing various substances available in the environment. Therefore the concentration of dioxygen stays extremely small.
On a planet with life forms that have developed catalysts (enzymes) for oxygenic phototrophy, like the blue-green algae (cyanobacteria) of Earth, the speed of producing dioxygen increases by many orders of magnitude. The consequence is that the ambient concentration of dioxygen increases until the speed by which it is consumed balances the production speed. The existing dioxygen is consumed by the oxidation of the magmatic rocks that are brought from higher depth to the Earth surface, by the respiration of a number of living beings that increases with the available amount of dioxygen, by fires and nowadays by the oxidation of various reduced substances, such as metals, which are produced by human industry.
In conclusion, life alters the speeds by which various chemical substances are produced or consumed, and this greatly alters the equilibrium concentrations of those substances in any environment where life exists.
What life does, is changing by many orders of magnitude the speed of certain chemical reactions, including the reaction by which water is decomposed, releasing free dioxygen.
The concentration of any substance in our environment is normally an equilibrium concentration, which is determined by the speeds of the chemical reactions that produce and that consume that substance.
On a planet without life, the speed by which the stellar light produces free dioxygen is very small in comparison with the speed at which the dioxygen is consumed by oxidizing various substances available in the environment. Therefore the concentration of dioxygen stays extremely small.
On a planet with life forms that have developed catalysts (enzymes) for oxygenic phototrophy, like the blue-green algae (cyanobacteria) of Earth, the speed of producing dioxygen increases by many orders of magnitude. The consequence is that the ambient concentration of dioxygen increases until the speed by which it is consumed balances the production speed. The existing dioxygen is consumed by the oxidation of the magmatic rocks that are brought from higher depth to the Earth surface, by the respiration of a number of living beings that increases with the available amount of dioxygen, by fires and nowadays by the oxidation of various reduced substances, such as metals, which are produced by human industry.
In conclusion, life alters the speeds by which various chemical substances are produced or consumed, and this greatly alters the equilibrium concentrations of those substances in any environment where life exists.