IMHO most episodes of highly episodic and non serialized TV series like TOS happen in alternate universes of their own. Some episodes are sequels to other episodes but most episodes happen in alternate universes different from the alternate universes of every other episode.
Think of the creators of an episodic series searching thousand and millions and billions of alternate universes and selecting the most suitable events from them to make into episodes.
We may note that "The Enterprise Incident" may not be a sequel to "Balance of Terror". It is not certain whether the surprising Romulan cloaking device is surprising since it has been improved since "balance of Terror" or surprising because the Romulans never used a cloaking device before.
In the latter case "Balance of Terror" would have never happened. Since it is already known in "The Enterprise Incident" that Romulans and Vulcans are related, we might suppose that "The Enterprise Incident" happens in an alternate universe where the Romulan leaders decided not to attack the Neutral Zone outposts but instead perfect the cloaking device and in the meantime open up normal relations with the Federation and sign the Nimbus III treaty from Star Trek V: the Final Frontier. No doubt settlers from Romulan subject races on Nimbus III would tell everyone that Romulans looked like Vulcans.
We could think that it doesn't matter if Kirk & Co. were correct to assume there would be no disastrous consequences of their actions in "The Enterprise Incident" because probably no later episode was a sequel to "The Enterprise Incident" and thus we don't get to see what may have happened as a result of "The Enterprise Incident".
As for why the Enterprise crew didn't worry about the risks of interstellar war or being disavowed by the Federation before crossing into Romulan Space, only Kirk and Spock knew about it beforehand and they were under orders from Starfleet Command. Kirk and Spock probably protested their orders when given and probably worried a lot abut what could happen. That probably helped Kirk to act like he was stressed out and incompetent before giving the order to cross into the Neutral zone because he would have been very stressed out.
Considering how foolish the plan seems and how unlikely to succeed, it seems probable that Kirk and Spock would have been more likely to disobey the admiral who gave them their orders and arrest him for giving illegal orders than to obey him. Unless there were some other factors not mentioned in the episode that made them more willing to take the risk.
Perhaps Kirk and Spock were informed that one or more persons in that particular Romulan fleet was a Federation agent, thus making the plan much more likely to succeed, for example.
Perhaps they were shown proof that the Romulans were planning war and that stealing a cloaking device was the only way to deter them.
Perhaps they were told that the Romulan Peace Party had suggested the mission to capture a cloaking device and discredit the Romulan war Faction.
But the other officers weren't told ahead of time and so the other officers and crew didn't know or worry about it until Kirk ordered them to cross into the Neutral Zone. And after the Romulan ships appeared the others were probably worried more about staying alive and escaping from the Romulans than the political and career implications and didn't talk much or any on screen about the long range fallout from the situation.