The problem is that JKR changed her mind about how the Fidellius charm worked. Originally, the charm was incredibly strong - only one person could ever reveal the secret. If that person died, no else would ever be able to know the secret. From the old JKRowling.com:
What happens to a secret when the Secret-Keeper dies?
I was surprised that this question won, because it is not the one that I'd have voted for… but hey, if this is what you want to know, this is what you want to know!
When a Secret-Keeper dies, their secret dies with them, or, to put it another way, the status of their secret will remain as it was at the moment of their death. Everybody in whom they confided will continue to know the hidden information, but nobody else.
Now, however, the official stance is exceedingly different. JKR changed her mind about the charm, and it is now significantly weaker:
The Fidelius Charm is not without its weaknesses. If the Secret Keeper wishes to do so, they may divulge the information at any time (although the secret cannot be forced, bewitched or tortured out of a Secret Keeper who does not wish to give up their secret; it must be given voluntarily). If the Secret Keeper dies, anyone to whom he or she has confided the information will become a Secret Keeper. This could involve many people, any of whom might be more willing to share the secret.
I say weaker because under the original rules, if you wanted to learn a secret, your only option was to convince the single Secret Keeper to divulge it. Now, however, if the Secret Keeper won't tell you, all you have to do is kill the Secret Keeper, and you suddenly have 20 new people who might be willing to tell you. Or they each might tell 20 people each. Kill the original 20 the first Secret Keeper told, and you have 400 people who could be convinced to tell you. For a wizard with no moral objection to committing murder, learning a secret protected with Fidellius is pretty easy.
Now given two passages in Deathy Hallows that refer to the protections on Grimmauld place, it seems likely that this change in the way the charm works came to JKR while she was in the middle of writing it. In chapter 9, we have the trio discussing using Grimmauld Place:
"We need a safe place to hide," said Ron. "Give us time to think things through."
"Grimmauld Place," said Harry.
The other two gaped.
"Don't be silly, Harry, Snape can get in there!"
Note that the only fear they had was that Snape could get in. Not that Snape had told the rest of the Death Eaters about it. They didn't worry that Snape had become a Secret Keeper, they only worried that Snape was one of the people Dumbledore had originally told.
This is despite the fact that 3 chapters earlier, they supposedly "knew" that Snape had become a Secret-Keeper:
They were often joined by Order members for dinner now, becaue the Burrow had replaced number twelve, Grimmauld Place as the headquarters. Mr. Weasely had explaind that after the death of Dumbledore, their Secret-Keeper, each of the people to whom Dumbledore had confided Grimmauld Place's location had become a Secret-Keeper in turn.
and not only that, that but they could merely hope Snape hadn't been able to reveal the location:
"But surely Snape will have told the Death eaters the address by now?" asked Harry
"Well, Mad-Eye set up a couple of curses against Snape in case he turns up there again. We hope they'll be strong enough both to keep him out and to bind his tongue if he tries to talk about the place, but we can't be sure. It would have been insane to keep using the place as headquarters now that its protection has become so shaky"
So we have one part of the book where it's a well-established fact that any Secret-Holder became a Secret-Keeper when Dumbledore died, and we have another part of the book where the characters act as though there's no chance Snape could have revealed the location to anyone. Combined with the change in how JKR originally envisioned the Charm working, it seems the only real explanation for the characters' motivations in that decision is, unfortunately, "Plot hole".