False, lmao.
Luffy had weaker haki than Katakuri, which was specified repeatedly in their mirror match type battle, and still had the correct proficiency to win in HIS specific fight barring that fact. He had worse FS and CoA, and still was able to win with Snakeman.
That alone proves that sword skill and haki output are different ingredients for a fight. You do not have to be a stronger haki user to be a better swordsman. I would 100% expect Mihawk to outclass anyone in swordsmanship. I do not expect Mihawk to directly win every single top tier fight if "Haki is King" as Kaido implies here.
He's never beaten Shanks lol, and he never beat Roger, so you can choke on that one for a while.