I'm not sure whether you're asking the reason no one betrays Nargothrond or the reason Turgon's policy seems overkill, but I have answers for both.
The main reason I think those who know where Nargothrond is do not reveal it is that there is no evidence of the bad guys learning of Nargothrond that way. As you said, it is the shift to a policy of open war led by Turin that reveals Nargothrond to Morgoth. He does not know of its exact location until this time, not though he and Sauron likely have plenty of Elves captive who might know the way. Moreover, I think getting information out of Elves who are determined to hide it is incredibly difficult. Gwindor didn't reveal the location when he was in Angband, and Edrahil and the other Elves captured with Finrod and Beren all chose to die rather than reveal who they were or where they were from. Even in SilmFilm, we have already shown Sauron needing to implement an elaborate plan to trick Edhellos into revealing the Kinslaying. Also, while I think Finrod would be more open about the location of Nargothrond than Turgon is about Gondolin, it's not like he is passing out flyers with the location of Nargothrond marked on a map. He probably shares it with the House of Finwe, high-ranking commanders, and other important people, but an ordinary Elven farmer or low-ranking soldier would not know the location.
I think the simple fact that Turgon is willing to kill other Elves demonstrates that he has gone to far. He is not placing faith in the defenses of Gondolin, in Ulmo, or his own kind's capacity to keep secrets, but rather allowing himself to be ruled by the fear of treason that was part of Mandos's doom, and, ironically, this is exactly what leads to Gondolin's fall through Maeglin.