Mihawk's goal has nothing to do with becoming Pirate King.
No one wants to become a Yonkou, they want to be the Pirate King but cannot. Aside from Whitebeard, this applies to all the Yonkou.
Shanks gave up the race when he placed his hopes onto Luffy.
Kaido, Big Mom, Blackbeard are still trying to get their hands on the Pirate King title. But we all know who will be the future PK.
Whitebeard did kind of achieve his goal of having a big family but he couldn't protect them till the end. Now a possible relic from his past is on the hunt for his family (Weevil and his mom).
In a world where the most powerful players fail to achieve their dreams, you have Mihawk, the man who's achieved his dream.
There are only 2 official World Strongest titles, he's got one of them and currently is the only one alive of the 2 holders.
In that world which "robs" people of their dreams, Mihawk is clearly one of a few rare winners.