My point is that we don't know whatever happened in their duel so arguing about that is of no use. What is important is that Oda considers Shanks a swordsman yet ranks Mihawk as the strongest swordsman.
Strongest creature
Strongest man
They all have amazing strength. They are top tiers.
But:
1) Shanks equally matched the strongest man, WB, in haki clash.
2) Shanks stopped Kaido from coming to MF.
3) Shanks dueled with Mihawk.
He is the only one who have clashed with these noble title holders.
So, it is reasonable to discuss the implications of "settle things".
You settle things with those who are fighting against you.
It has to be the start of discussion if we are discussing Shanks vs Mihawk.
I have given enough proof that title holders can be challeneged (Like Zoro vs Kaido, Luffy vs Kaido).
However, Shanks equally matched previous 2 title holders, i.e., WB and Kaido.
Plus, the card here says:
"A corner of the "Four Emperors", the sea castle that reigns over the New World. A master swordsman who fought on equal terms with the world's strongest swordsman, Mihawk. In addition, his terrifying 'hao-colored aura' stunned the stalwart Whitebeard Pirates one after another"
So, Shanks also fought against Mihawks on equal terms.
Hence, when it comes to 1v1 fights, Shanks can be equal to Mihawk despite being an inferior swordsman.