Where oh where do I begin... First with Twilight... No I didn't read it... well I tried and couldn't make it past the horrible writing. So I made it 5 chapters before stopping. Watched a few riffs of the movies and got the basic story... Horrible point of comparison. Seriously... The secrets being held in Twilight are nothing like the secret in DC. It is a poor point of comparison...
Honestly, I see it closer to Sherlock Holmes post-Reichenbach Falls if ANY literary parallel must be made. The main detective somehow miraculously survived a brush with death with one of the main forces in a large lethal organization so much so that he had to ensure that everyone thought him dead and basically stay underground with one person knowing the secret who could handle his assets. The only difference? The time against Moriarty's organization was only implied and thus didn't last nearly as long, preventing others from discovering. But I personally can imagine Gosho writing DC as an homage to Sherlock Holmes during his time against Moriarty's Org with a few twists of his own (shrunken body, and basically a first lieutenant doing the killing not the leader, and the bad guy surviving leaving more of a challenge for the protagonist)
Now, you made an interesting choice of words... "Reckless." Shinichi was ALWAYS reckless. "Hm, a guy who looks like a professional killer and his partner... Maybe I should follow the partner completely forgetting about the killer looking dude." Reckless? Lethally so. It is pure dumb luck that Shinichi did not die from a poison that kills the majority of its victims. I'm pretty sure that he did more stupid stuff in the past too that we don't know about, due to his arrogance and overwhelming idealism.
Now moving on to the point of this question... Revealing his identity. I shall permit the notion that initially, when he knew nothing of the BO and was basically running blind before Haibara and Akemi, he was totally correct in his assertion, because he didn't know anything either. However, after that point his decision to persist in his ruse to keep his identity secret became increasingly reckless. And ironically, he became reckless in actually KEEPING it secret.
First, not allowing his father to get Interpol involved. Why? Because he didn't want to go into protective custody because of Ran. However, he could have told his father to just TIP THEM OFF AND GET THEM INVOLVED without mentioning him. That way he would have his father and at least a few ICPO agents helping uncover the Org, along with the FBI who was apparently already on the case.
Second, allowing pretty much anyone with half a brain to figure out his secret. Ran is apparently the most gullible person in the series, because Heiji figured it out like it was nothing and he basically allowed Takagi and Sato have extreme suspicions. In fact, he basically TOLD Takagi that he was not normal in the elevator. There is no way Takagi forgot that or wrote it off as childs play. He is pretty reckless with keeping his own identity a secret... except....
Third, THE FBI! I'm putting this up with extreme arrogance and recklessness. Hey, Shinichi... There are professional agents whom have YEARS of experience dealing with organized crime syndicates in the USA... I think they are called the FBI, perhaps you've heard of them? Yeah, THEY KNOW WHAT THEY ARE DOING, YOU DO NOT! You should have told them once you realized they could be trusted. Why in blue blazes have you NOT DONE SO ALREADY?! Do you REALLY think that you can defeat the BO by basically using them? Really? I think you need to think again, since... you know... you nearly got killed by them a few other times. And nearly got Kogorou and Ran killed too. In fact, had it NOT been for those professional agents known as the FBI, THEY'D BE DEAD ALREADY (and Vermouth telling Gin as well)! Probably you too! AIN'T THAT GRAND?! Stop being a reckless idiot and TELL THEM ALREADY!
Fourth, Ran and Kogorou... Less reckless more stupid. Really, as Madeline said, this is a case where ignorance is NOT bliss. I mean considering that they have been targeted already... And the only reason they aren't being watched as closely is because of Vermouth... Yeah... He should have told at least Ran right after that incident. Doing so would not have changed status quo so much. Ran would be conflicted because she'd realize that it was Shinichi solving all those cases and not her father. But in the end, I think she would believe in him enough to keep things the way they are. However, she would also know to be more careful. Possibly avoiding situations like the one where she nearly got killed by Calvados had it not been for Vermouth stopping him. That and his apparent horrible aim (I'm pretty certain that BO gunmen are trained to shoot moving targets)... She would still want to help, but if he told her that the FBI were already helping, there would be no reason for her to go to the MPD (which would be the only other possibility other than telling her father). She might become a bit more proactive, but that is about it...
So in the end... Yes, Shinichi was, is and probably always will be extremely reckless and arrogant. I sincerely hope SOMETHING happens that will FINALLY get him to change that. Personally I think either Ran or Kogorou ending up in the hospital because of the BO would do that... Or something else... I don't know... But I do hope it happens...