None of the criteria there would be satisfied. With them detained outside the house and not allowed to return inside, they can't harm the victim, escape, or destroy evidence.

In the eyes of the law, does a missing child automatically fulfill the "imminent danger" or "imminent destruction of evidence" parts of exigent circumstance? Or does there need to be some other evidence to support those reasons?

Further, does the law, as written, allow detainment of individuals while those reasons are decided?

Until someone can describe the legal link between the fact it's a missing child and the conclusion these are exigent circumstances from a legal perspective, I can only interpret the pointing out that it's a missing child as a plea to emotion at best and a thought terminating cliche at worst.