That assumption is built into the original question. If the information is immediately shared with the players to use as they see fit, the original question is nonsensical.This is well presented, but it has at the heart of it the assumption that the DM should keep the information about the monster secret as a matter of course. I'm not sure that's entirely warranted, although it is, largely, how the game has been played for quite some time. I'm going to challenge that idea, though.
A quality encounter is one where the players are engaged in the action and make choices that have heft -- that matter and change the fiction. This doesn't happen very well if the DM's running gotchas on the monsters because the players are playing guessing games to figure out the secret or asking for rolls to know things. To me, this represents a failure on the part of the DM to present a complete situation where the characters can act. The DM should have been foreshadowing the threat, and the nature of the threat, in the scene framing or in previous scenes. So long as the DM provides strong avenues to discover the secrets of the encounter, then it becomes a meaningful player choice if they fail to follow up. It's in the keeping of the secret, or gating behind random chance, that this is lost.
So, to make the case for Yes, the DM should endeavor to provide the secret in play such that the players can make decisions with heft. Doing otherwise is just playing gotcha.
I've been doing this for a few years now -- providing lots of info, often for free. There were some growing pains where I felt that I was making encounters too easy, but that passed quickly for two reasons -- one, I stopped building encounters based on the gotcha so that wasn't a problem and b) I found out that I could give my players my notes straight out and they'll still find ways to screw it all up by the numbers.