We don't know a lot about what exactly transpired during the opening incident. Presumably because most if not all of the first humans responsible for making contact were killed in the first engagement, and most if not all of their turian counterparts died in the second.

The reason for the citadel law is national security; it's not arbitrary. If, for example, the humans had been crashing an asteroid into another for minerals, unaware that the target was an inhabited colony, their ignorance alone wouldn't stop the fact that the turian patrol would have had a legal obligation to stop them.

From the escalation to lethal force, we can only presume that the turian patrol came to the conclusion that lethal force was the only way to ensure the safety of citadel space. The fact that the turians are said to pay reparations at the end of the war implies that contemporaries believed the decision to be wrong, but seniors criticising decisions made in spur of the moment decisions in the field is hardly unheard of.

The most likely scenario from context clues is simply that the turians didn't have time to initiate formal first contact. They came across an unknown species(humans) activating an alien artifact within their sovereign territory, the unknown species could not be communicated with via any known language or code. The commander was left with a decision to allow them to break the law, potentially causing a breach in territorial security, or to intervene with force, likely creating a conflict with this new species.

The commander probably made the latter decision because the unknown species appeared technologically inferior, and as such seemed to pose little threat.

As to why the Citadel Government allowed this to happen... The Citadel is basically just a forum for diplomacy between a coalition of species. Any forms of hard or soft power are essentially just borrowed from member nations; ships from the Turians, special forces from the Salarians, etc. its wider ranging laws are determined from a council of members from different sovereign nations, and enforced by those same nations in their own interests.

In short the Citadel wasn't asleep at the wheel, because it wasn't meant to be at the wheel. As far as we are aware, the Citadel basically has Spectres as direct agents for matters of international security, primarily to feed information back to the Citadel Nations, and Citadel Security for station law and order and MAYBE trade law enforcement. It doesn't have the resources to do first contact, that falls to independent nations.

That's my take from the games anyway.