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Because there is no reason to prove anything. Mihawk is portrayed as standing at the top and Zoro himself acknowledges that. It is also in-character for him to not do unnecessary fighting. He chased Don Krieg but let him off the hook immediately after enjoying himself with Zoro, not even...

Mihawk and Shanks were rivals, confirmed to be equal, honing their abilities while fighting each other ( = same growth rate). By losing his arm, Shanks sacrificed strength. Their rivalry became obsolete, as Shanks was not equal to Mihawk anymore. And Mihawk is capable of gauging someone's power...

Mihawk and Shanks had duels that Whitebeard remembers as one of the legends. Being rivals without surpassing one another, it also means that their growth rate was roughly the same. So Shanks, after being nerfed through losing his arm, doesn't suddenly get a boost in growth rate so big it...

Roger was taking a bath without discomfort. While Luffy after stretching his arm out under water was greatly taxed.

It's probably because the WSS is a title swordsmen actively pursue by challenging other swordsmen they view as powerful too. With Zoro's goal being Mihawk and not Shanks, who already goes toe to toe with Kaido. Zoro's goal would be questionable if it was just due to his perception of Mihawk...

Not really. You are the WSS if you are considered the strongest and not if you are interested in contesting for it. Whitebeard was not interested in the WSM title either, yet he was the WSM. Mihawk is interested in fighting someone equal or greater. If he heard of a potentially stronger...

King is a mediocre swordsman with powerful heritage and a useful fruit. While the strongest people in the verse are top tier swordsmen, even "pure" ones without a DF; Shanks (no DF) Oden (no DF) Mihawk (no DF) Roger (no DF) Rayleigh (no DF) Powerful swordsmen with side powers also fall under...

The strongest people in the verse fall under the WSS debate, which is enough.