If we actually pay attention to subtleties in manga, even before the timeskip, the title of World’s Strongest Swordsman has consistently been placed on a higher pedestal than the title of Emperor of the Sea and Chapter 1058 reinforced that idea in multiple ways.
To set the stage we have to go to Chapter 1033 – There are only two confirmed Kingly Ambitions: Pirate King and World’s Strongest Swordsman.
Well this isn’t the first time Mihawk’s and Roger’s titles have been paralleled, one of the very first chapters of Manga setting the stage for what is to come has Zoro and Luffy’s ambitions outlined in the title of a chapter.

It also needs to be noted that Zoro was substitute for Luffy’s head against Kuma when Zoro told Kuma he will one day become strongest swordsman in the world. That is surely a sufficient replacement, aka head of World’s Strongest Swordsman is enough to consider sparing head of Pirate King.

“You have such an Ambition”: Said Mihawk’s fellow Warlord and really Dragon’s former right hand man. And obviously Kuma is willing to take Zoro’s head.
Now we know what the perception of title of World’s Strongest Swordsman is to a Warlord in comparison to Pirate King title. A title holder whose head is sufficient replacement for for Pirate King’s title.

It’s only fair we must also look at how a Warlord or in this case former Warlord perceives Yonko title in comparison to Pirate King title.
In the hindsight this might seem like just a diss on Big Mom from Jinbe but it runs deeper than that, because that is my real point. Emperor title is obviously worth something just like rank of Admiral is worth something, just like being a Warlord is worth something. But it isn’t given the level of respect the title of World’s Strongest Swordsman is given.

Again remember I am just talking about title in this instance not necessarily title holder because Whitebeard is obviously much more than just an Emperor, he is essentially King of an Era.

But Chapter 1058 shows World’s Strongest title is worth more than Yonko title, as a matter of fact so much more that despite World’s Strongest Swordsman supposedly working under an Emperor, we are told as such:
Fitting Bounty for Emperor and Mihawk’s Captain is 3.19 Billion.
Fitting Bounty for Strongest Swordsman in the World is 3.6 Billion.

So World’s Strongest Swordsman is not only given the distinction of Kingly Ambition, a head that is worthy replacement for the head of Pirate King, but also we are straight up told by Brannew that World’s Strongest Swordsman is worth more than an Emperor.

Gotta understand the technicalities too – An Emperor has to be an Emperor for Multiple years, do some crazy things, have some kinda insane pedigree like Rocks Pirates Crew, a Crazy Strong Crew and then the title is worth something.

Considering Blackbeard at the 2,247,000,000 Berries can become an Emperor, minimum cut out for the worth of an Emperor is extremely Low.

This isn’t necessarily saying Mihawk is above every Emperor ever, that is a thread for the future, as there is someone like Whitebeard who is obviously far greater an entity than your average Emperor, he is essentially King of an Era and certainly the worthy rival of Pirate King not to mention Strongest Man in the World in his prime.

But why would a man who is a King of Swordsman, a title that is someone’s kingly ambiton ever go for the title of mere Emperor.