Our assumptions are based on the fact that Mihawk has not shown haki at all. The second Mihawk shows haki even a little we will agree with you lol.
You forget that when mihawk asked what luffy's goal was he said that in order to become pirate king luffy would have to become stronger than him. Which means that all the yonkou's who are at the top of the pirate world are stronger than mihawk. And since shanks is a swordsman with one hand and is stronger than the greatest swordsman in the world we can only assume that this is because mihawk does not have haki.
Mihawk agreed to teach Zoro the way of the sword, he never agreed to teach him haki. Whether mihawk has haki or not does not change this fact and mihawk is not one to give out gifts for free.
12 years ago, Shanks and Mihawk were rival. When Red Hair saved Luffy from the sea king, he proved that he could use it perfectly (conqueror). We know that Shanks and Mihawk were equals and rivals at that time and Haki was mandatory for a draw. When Mihawk visited the Yonku, both did not even seem intimidated by each other. IMO, they are still equal.
I think Yonku is not only individual strenght but more importantly strong allies and flawless team work. When Mihawk and Kizaru attacked WB, the Yonku did not even move a muscle because he knew Jozu and Marco would stop them. He had total faith in them. It is more about tramwork than individula strenght.
When Luffy unleashed the "conqueror" during Marineford, Mihawk was not even surprised and he knew what it was.