Another way to show this is through the non-existence of perpetual motion, the Feynman way. As he says, we first assume that perpetual motion and hence the creation of energy is not possible. Next, let engine $A$ be a reversible engine working between temperatures $T_1$ and $T_2$ where $T_1>T_2$ and let it absorb heat $Q_1$ frome the reservoir at $T_1$ and give out heat $Q_2$ at reservoir at temperature $T_2$ doing work $W=Q_1-Q_2$ in the process.
Consider another reversible engine $B$ working between the same temperature but having different efficiency. On taking heat $Q_1$ from the hotter reservoir, let us suppose it does work $W'>W$ and therefore gives out heat $Q'$ at the colder reservoir which is less than $Q_2$. After operating engine $B$ for one cycle, we could use $A$ in reverse to siphon out $Q_2$ from colder reservoir, submit heat $Q_1$ at the hotter reservoir with a work input of $W$. But since $W'>W$, therefore the net result of oprating $B$ followed by reverse $A$ will be to have drawn heat $Q_2-Q'$ from the colder reservoir and completely converted it into work $W'-W$, without any other change or entropy since the engines are reversible. But this contradicts the well know statement of second law of thermodynamics:-
Clausius staement-"Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time."
This argument shows that working between any two temperatures, irrespective of the nature of the engine, a reversible engine shows the maximum efficiency and all reversible engines must show the same efficiency (Carnot principle) to not violate second law of thermodynamics. This idea, originally Carnot's proves that all reversible engines have the same efficiency as a Carnot engine. Your argument uses certain assumption about the working of a reversible engine which might not be universally true.