The idea of the Valar "withholding critical information" from Gil-galad implies that they are allies within a shared conflict. That's not really how things worked. Even the war vs Morgoth wasn't really participated in by the Valar. A few Maiar joined the fray, or at least lived in Middle-earth (e.g. Melian), but it was a war between the Elves and their allies and Morgoth, and one that wasn't particularly approved of by the Valar, at least initially. The Valar only interceded at the very end.
This is even more true of the Second Age and later. The Valar helped combat Sauron through sending the Istari, but that's pretty much it. We can speculate on whether they would have intervened again if Sauron won the War of the Ring, but it is just speculation.
I think the reason the Valar didn't intervene is related to why Gandalf and Galadriel both refused the One Ring: core to Tolkien's philosophy is that evil and control over others and their fate are synonymous, or at least isomorphic. Elves and Humans are free people and intended to make their own way. Thus, the Valar kept out of things, for the most part. Thus G&G refused the ring, because they knew that even if they used it for good, the very act of using it--as power over others--was evil.
I mean, there's no reason that the Valar couldn't have jumped in at any time and found Sauron and brought him back to Valinor in chains. There's no way Sauron could have withstood their combined might. Even at his most powerful, he probably wasn't equal to Morgoth (except maybe when he had the One Ring); plus, Morgoth had dragons and Balrogs. But they didn't, because the Valar were servants of Iluvatar, and essentially told not to intervene in the fates of Iluvatar's children (Elves and Men).
The point being, the Valar did not directly engage in the conflicts of Middle-earth--and barely even indirectly, and only through intermediaries. They weren't "headquarters" for the good guys; they were largely above/beyond/outside the various conflicts.