Fans still don't understand something, there are three things we know about Mihawk
1. He fought Shanks
2. He didn't fight other Top Tiers
3. He is waiting for Zoro (And Luffy) to surpass Shanks
It's not that difficult really, these three Info reveal everything
Shanks is Strongest Characters & he & Mihawk are Rivals, but since Shanks lost his Arm, Mihawk lost interest in fighting him & is waiting for someone New
In other words Mihawk believe Shanks is Greater than all other Yonko & Top Tiers
He have no interest in Relics of Old Era
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As for how Oda is handling him, it's also very simple & portrays Mihawk as Greatest alongside Shanks:
1. Just when Story started, we were introduced to Three Dangers of being a Pirate: Navigating the Unforgiving Seas, Meeting powerful Foes & surviving the Military Power of Pirates (Crews) & Luffy learned that he must Get Stronger, get Powerful Crew & survive the Ocean with Great Ship & Navigator, but guess what? Oda randomly introduces this Character who roams the Oceans in literally a Chair, destroys Fleets of 50 Ships because they disturbed his sleep & he is World's Greatest Swordsman & also chases Pirates through Grand Line & also one of Blues just because he was Bored! It's like everything that Oda was hyping up about Luffy's Journey & Dangers of the Sea, this guy throws them in the Trash Bin
2. Oda does Hype almost every single Character when they appear, but that Hype lasts for some period, however after 1000 Chapters, Mihawk is still unexplored in the Story. He is clearly an Endgame Character & Oda decided to Hype/Introduce him decades before which means he is among few Characters that Oda want us to always wait for his time to shine, otherwise he could have just introduced him shortly before Final Arc similar to how he introduced Kaido & BM couple Arcs before their own. This is something that Oda only did with Shanks, Mihawk & Dragon. They are clearly the Characters with Longest Hype & the most waited for
3. IMO Shanks is hyped beyond any other Character in the Series, so when we see Oda sacrificing some of that Hype for the sake of Mihawk, it tells so much about how Incredible Mihawk is. Teach Hyped Shanks, Kaido hyped Shanks, WB hyped Shanks, Roger hyped Shanks, Oda himself hypes the guy, but when Mihawk is involved (Who is almost same Age as Shanks & same Era), Oda decided to grant Title of Greatest Swordsman to Mihawk instead of Shanks, decided to make Mihawk the one who refuses fighting Shanks & bad mouth him instead of opposite, decided to make WB mention Mihawk when he meets Shanks even though he didn't need to at all & also the Latter says that his eye injury isn't from Mihawk (Implying that Mihawk was one of the guesses of WB)
4. During the Summit War, Oda had to handle so many Powerful Characters & they all needed their moments to shine (Most at least), and we saw many Characters Interactions, however Mihawk is the "Only" Character in that Top Tier War who wasn't interested in anyone except Luffy, how can you be inside such a Huge War alongside most of the Greatest Figures in the World & your interest is mainly focused on the only Rookie among them! That shows how much Mihawk truly isn't interested in Old Era or Current One.
We had few Scenes of him with WB, Jozu, Vista, Croco ... etc but even though he showed respect to them, he didn't continue his interest in them, but with Luffy? He chased him many times & multiple Characters had to step in
5. In SBS, Oda said "Legends among Men like WB, Shanks & Mihawk" showing that Oda himself group Mihawk with those two even though he is neither Old not have Big Influence like Yonko, he is a Young Lone Fighter & yet Oda calls him a Legend with WB.
Viver Card further hype him even more claiming that he is waiting for someone even stronger than Shanks & that he sit at the top of the world & is greatest Swordsman in both name & actually. And even in earlier Databooks, it was stated that Mihawk is even stronger than Shanks
Mihawk is Hyped in every way, His Introduction + SBS & Databooks + Shanks Parallels & Portrayal + His Title + His Personality & relation to Luffy/Zoro + His reserved spot in Final Arc + Him being only Top Tier from Current Era alongside Shanks, all of these show that Mihawk & Shanks are Greatest Characters currently in One Piece similar to Roger & WB in the Past
So as i said, Mihawk never got interest in Yonko or any +30 year old Character, because he already have Shanks