No the problem is you guys get way too caught up on fanmade definitions of swordsman to make up headcanon.
Pica clearly wasn't just a swordsman either lol. Carrying a sword does not mean you are exclusively competing to become the WSS or can exclusively fight that way.
This is why Shanks vs Mihawk discussions are so heated, because 1 side can't understand the concept of a duel potentially being based on exclusively using swords, and not entirely based on the complete skillset of someone specifically.
There's nothing saying Shanks can't just drop Gryphon and fight a different way. Hell, there's nothing stating Mihawk can't either. In terms of SWORD TECHNIQUE, Mihawk is the best. In terms of anything else, it's not defined at all in this series whatsoever. Nothing actually states Mihawk is stronger than Shanks lol. As a sword user, sure, but that's ONE avenue of fighting, which hasn't even been fully defined for Shanks
And THATS why this interaction with King is so important, because the story is literally telling you the concept of "swordman" does not matter to people like King.