not proof, your assumption - disproved by the fact that Rayleigh used zero named attacks vs Kizaru. Do you believe that all his strongest attack could do is inflict a paper cut?
If you don't believe Rayleigh used his strongest attacks vs Kizaru (which he didn't as we've seen zero named Rayleigh attacks), then your statement that Rayleigh is doing everything in his power is absolutely false.
You cannot prove Rayleigh used ACoC vs Kizaru so any statement claiming Kizaru is somehow capable of using ACoC because Rayleigh did is completely baseless. Furthermore, any statement claiming Kizaru even posseses CoC at all is completely baseless. Finally, we've seen characters be equally matched in AP despite using different abilities in Kuzan vs Garp so any statement claiming Kizaru can use any ability Rayleigh's using simply because they clashed is so unbelievably baseless that it's utterly laughable.