Wrong.
It wasn't damage but the fact Luffy obtained the ability to damage him in a short amount of time. In less than 3 weeks Luffy went from being unable to damage Kaido to damaging him.
That was also the reason Kaido ended up being very disappointed in Luffy later on. He got too excited over Luffy's growth.
Another person obsessed with knocking Kaido down. That has amounted to little to nothing. It won't be enough to beat or kill Kaido.
Zoro attack lead to Kaido breathing heavy.
Zoro was in poor condition so that means if he was in better condition he would have hurt Kaido more and fought him longer. Strongest attack at 20% is still an attack at 20%.
Oda had Zoro fight Kaido first to show us he had the power to cut, scar and fight Kaido at the beginning of the war. His purpose to fighting King now is for Zoro to gain the power to kill Kaido.