If the information that Dr. Kovich wasn't revealing to Burnham actually would increase the danger if she knew. I don't see that her knowing about the progenitors and the Romulan doctor increases the danger.
I suppose there could be more to it that we'll find out. I've only seen the first episode. But when it was revealed, I wasn't like, oh yeah, I can see why he didn't want that getting out because that information itself increases the danger.
But, like I said, didn't reduce my enjoyment, just made me shrug. I suppose bureaucracy and all hasn't changed by the 32nd century!