Maybe LE is okay legally with the handling of the interview at Uninversal, but I personally think (JMO) that as soon as they confirmed the lie that she did not work there, they had her in their sights as the suspect, as they would have to be blind not to do so. I don't believe they felt they had a missing child at that point, and YM's statements to her about the child being in the trash can, etc. support the idea that he felt he had the killer right there in that room.
But I hope they did it right, from a legal standpoint.
I believe they did-KC was not charged with murder for months. Weeks that it took for the police to come anywhere close to presenting their findings to the state, weeks before the state made a decision what to do about it.
In the meantime, Yuri, the other detectives, and the FBI searched for a living Caylee. The FBI went door-to-door asking people to present their children for a visual identification.
I believe JB wanted to point out that LE knew she was lying to them prior to Univesral and were going to charge her with that, so she should have been Mirandized there. But LE could not have known whether they were lies in KC's mind (though they were lies in reality), or whether KC was scared & confused, on medication or mentally unstable-until they spoke with her further. She did not simply say "I was not there," she made up elaborate stories. They had to want to know if she was just looney tunes. They found out for sure during the Universal interview that she
meant to lie in such a grand fashion.
As to YM's statements, I am not sure YM was
sure. I believe he held out the possibility that Caylee died accidentally and KC was afraid to tell anyone. It is thought by some that they did not Mirandize her so that she could further incriminate herself, but I give greater weight to the argument that they did not Mirandize because they wanted to find Caylee, they wanted KC to feel free to tell them, and then finally, they tried a scare tactic.
Once all of those things were exhausted during the Universal interview, I agree with you that they pretty much knew. But they fell back, regrouped, surely talked to Lawson Lamar, and proceeded to arrest her.