It's not a trial, it's the public asking their government to tell them what's going on in this matter. And the government flatly refused. From this, it's absolutely reasonable for the public to think its government is hiding something now. And it's an order to _others_ in the DoD to remain silent. If it were a trial, this is ordering potential witnesses to remain silent on threat of termination.
What the White House is saying by claiming presidential privileged is 1) the president did indeed discuss it, which in itself is problematic and 2) that discussion is on par with a national security issue, which is an additional abuse of power.
As an aside, the 5th exists to prevent coerced confessions. In court, it is not to be taken as a sign of guilt because if it were, the point of the law would be moot. Where no court (or police) are involved, there is no possibility of coerced confession. Which is why everyone correctly views refusing to answer questions to friends, parents or reporters as very much a sign of guilt. And why confessions signed in police stations are very suspicious in many people's eyes. It certainly does not allow a government to refuse to answer to its people.