Clausius' statement about heat not being able to flow spontaneously from a cold body to a warm body is sufficient to prove that no engine can have an efficiency greater than that of a perfectly reversible engine. But it's not enough to prove that the Carnot engine is the only reversible engine. For example, there could be a perfectly reversible engine where the gas at initial volume V1 and temp T1 draws heat from a hot reservoir, expands doing work against constant pressure till the gas reaches a higher temp T2, then makes contact with a cold reservoir to contract isochorically to a lower temp T3 and lower pressure P2, then makes contact with an even colder reservoir to bring down the temp isobarically until the volume contracts to the original volume V1, and then makes contact with a hot reservoir to heat up isochorically back to temp T1 (the original state). This seems to be as reversible an engine as the Carnot engine. The only difference is that here more than two heat reservoirs are involved. In fact, a Carnot engine is the only possible engine which can operate between just two temperatures; all other engines require one or more heat source or heat sink at some intermediate temps. This has a bearing on the issue and make the two not comparable. It can be graphically demonstrated that, between the two outermost temps (ie between the maximum and minimum temps), the Carnot engine is the most efficient - see this excellent video.https://www.youtube.com/watch?v=6-6kXurDUUc

Clausius' statement about heat not being able to flow spontaneously from a cold body to a warm body is sufficient to prove that no engine can have an efficiency greater than that of a perfectly reversible engine. But it's not enough to prove that the Carnot engine is the only reversible engine. For example, there could be a perfectly reversible engine where the gas at initial volume $V_1$ and temperature $T_1$ draws heat from a hot reservoir, expands doing work against constant pressure till the gas reaches a higher temp $T_2$, then makes contact with a cold reservoir to contract isochorically to a lower temperature $T_3$ and lower pressure $P_2$, then makes contact with an even colder reservoir to bring down the temperature isobarically until the volume contracts to the original volume $V_1$, and then makes contact with a hot reservoir to heat up isochorically back to temperature $T_1$ (the original state). This seems to be as reversible an engine as the Carnot engine. The only difference is that here more than two heat reservoirs are involved. In fact, a Carnot engine is the only possible engine which can operate between just two temperatures; all other engines require one or more heat source or heat sink at some intermediate temperatures. This has a bearing on the issue and make the two not comparable. It can be graphically demonstrated that, between the two outermost temps (i.e. between the maximum and minimum temperatures), the Carnot engine is the most efficient - see this excellent video.