If you think about it, Vector being the only adult of the main cast makes you think how much different he is from the rest. Vector may act childish sometimes but has control of his own emotions and duties than Charmy and Espio. Charmy is a child with hyperactive personality while Espio may be a serious ninja as the voice of reason but he still a teenager dealing with problems that any teenagers would feel. Charmy and Espio has express their problems and issues about the matter but Vector hid it well since he has to be the adult who has to be by the younger cast side. If Vector didn't hid it well, then the Chaotix would have been more chaotic.
Vector is such a fun and interesting character to me. Sure he's a goofball, dork and silly guy, but he is so much more. He's also noble, kindhearted, good leader, and a good friend.
He's always taking care of the people in need and his teammates. And being the adult of the group he tried his best to be the best leader he can be for his boy's. I see Vector in struggling going into adulthood, which all us 18-20 can relate (I sure do–). He's is a adult but his still young and there still so much left for him to learn. But while being young he is still an adult. Vector needs to be matured and responsible for Charmy and Espio. He needs to put on the role of the responsible adult in any situation because he is the adult. And I think Vector having that mindset, scares him if he ever fails as a leader.
I wanna showcase the Chaotix Casefile last scene in episode 4, where he tells Espio: "I'm the one who accepted a job under false pretenses! How's that going to play if the rest of the private investigator community finds out about it?" He's basically saying: "I should have know better because I am supposed to be the adult." Especially when then he says: "Because right now am an abject failure!"
Vector is blaming himself for have not known better and taking the job that was lead by false pretenses, and then he adds how the private investigator community is going to reacted about it. Probably adding the shaming Vector for not doing good enough. While Vector can keep his emotions in order and keep a mature conversation, there's no dout that feels a lot of pressure in trying to maintain the adult role for his boy's.