Yes Mihawk is stronger regardless of what people think. I don't think the gap is big though judging by their countless duels and the fact it was unsettled, but considering it was their main fights and that Mihawk is now renowned as the strongest, Mihawk may have won their last duel
Strongest Swordsman > Other swordsmen
And Shanks is depicted as a swordsman who even had/have a rivalry with Mihawk and had countless duels, which is a battle where 2 people use similar weapons(swords in this case), it's also where his greatest accomplishment come from(it's the main thing Whitebeard remembers)
In before Shanks can kick too, as long as swordsmanship is your main fighting style, you're a swordsman and Mihawk will defect you
That's why Ohm who was a martial artist was considered a swordsman because swordsmanship was his main fighting style despite being good at martial arts too. That's why Kaku was a swordsman despite having a devil fruit and being a rokushiki user, because his best was in swordsmanship
In before Haki is a fighting style, no it's not. Garp(armament master) is known for his brawling fighting style through his fist(Garp the fist), Chinjao(CoC and Armament user) is known for his drill(fighting style and destructive power), Ohm(CoO specialist) is known as a swordsman, Vergo(Armament specialist) is known as a bamboo master and so on and so on. Haki has never been a fighting style. It's a supplement in your actual style of fighting. So as long as Shanks will be depicted as main swordsman, he'd be below Mihawk