@Rosella.Fiamingo 's thread got me thinking of this particular issue.
In case of Mihawk and prestige of his title, there is a much easier explanation
Combat related ambitions are often given lack of focus, such as not a single soul wanting to go after WB's title directly.
But it is by default because of scope of such titles.
Like you don't need to be gunning for WB to be WSM, you need to be strongest to be WSM. So if you are a fighter and you are getting stronger, and trying to be as strong as you possibly can be, innate goal has to be strongest in the world.
If you are practicing swordsmanship as a kid like Shanks has, innate goal has to be Strongest swordsman in the world. Those people in the Dojo Zoro is from, are trying their absolute best to be greatest swordsmen they can possibly be. There are countless Dojos like those where there are people training to make themselves as good a swordsman as possible.
That automatically gives Mihawk's title prestige, as its not a judges contested title, there is no one way to become WSS, you don't need to beat Mihawk to be WSS, you need to be WSS to be WSS meaning you just need to be stronger than every other swordsmen.
In the contrary a title like Pirate King requires specific efforts, you can't just become Pirate King by chance. Even if you are strong enough to be pirate king, talented enough to be pirate king, have leadership skills of Pirate King, you won't be pirate king until you get the poneglyphs and go to Raftel.
Yonkou title is of similar nature to Pirate King title where you can be talented enough to be an emperor like Teach was prior to his battle with Marco and the rest but he only became an emperor after winning that war. Being a Yonkou is a positioning in the World which you need to make specific efforts directed towards specific goal to get, its more of a doing the check list rather than having a certain amount of talent and skill that is requirement for Whitebeard and Mihawk's title.
Then how come no one, not a single soul, no Worst Gen member, no Gifter, no Pleasure, no pirate in East Blue wishes to be an emperor?
How come Emperors themselves don't wish to be emperors?