Conversely, while other captains may have had careers as interesting as Kirk's, we should be safe to assume no one else in his time had careers that were more interesting than Kirk's. That would imply we haven't been getting the most interesting stories possible for 50+ years. So Kirk has to be one of the very best. Otherwise, what's the point of telling his story vs. a better, more accomplished captain, whose feats would presumably make even better stories?
The counterargument there is the existence of spinoffs. Lots of successful shows spin off additional shows about other characters having their own separate adventures at the same time. TNG produced DS9 and VGR. The Six Million Dollar Man spun off The Bionic Woman. Hercules spun off Xena -- and was overshadowed by it. Law and Order and CSI each had multiple spinoffs. Doctor Who had Torchwood, The Sarah Jane Adventures, etc. Arrow has spawned a whole universe of superhero shows.
After all, it's not like this is really happening in some parallel reality and we're just observing it. What happens in a fictional universe is what the writers make up. So there are as many equally interesting, equally accomplished casts of characters to tell stories about as there are writers willing/getting paid to tell them. The only reason that Kirk is the only 23rd-century captain anyone in later shows talks about is because he was the only one who had his own series. But I guarantee you that any future Prime-universe shows set post-TOS are going to mention the Shenzhou and the Discovery alongside the Enterprise as important 23rd-century ships, because they're now there to be talked about.