Whys it a necessity for a bunch of people to care? I think it’s a personal goal for Zoro and we don’t even know if Mihawk cared about having a title, he just became famous for what he does and it stands out in the story only really because it’s Zoro’s dream to be the best around as a sword fighter. What we do know is that Mihawk sought out challenging opponents to a point where he felt like there’s none left who can give him the thrill he seeks in battle. Even Vista said all swordsmen naturally want to challenge Mihawk, to test themselves like on a personal level. That’s how I look at it, it’s not like pirate king where there’s a bunch of people racing and competing for that throne. Otherwise I’m sure there would have been several prominent characters going for it and Zoro might have even had a rival character to compete with throughout the story but it doesn’t seem to be the case.
imo, either you have a worthy rival or you dont beg the title holder to train you, thats it.