Did Snape really betray Dumbledore

Why didn't Snape 12 reveal Grimmauld Place after Dumbledore died?

This is just one speculation from me and I have nothing to prove:

Maybe he did it disclosed.

After Dumbledore's death, the order was temporarily dismantled and relocated from Grimmauld Place 12. Snape could have revealed the OoP headquarters to Voldemort, with him, Snape, and a few other Death Eaters visiting. But this revelation could happen after this Snape knew that the Order had completely removed anything that could be used against them (perhaps through Mundugus we know that he used the Imperius Curse on him). Voldemort found nothing after their visit and left the place.

As for the taboo curse, if we take into account that this curse could not break the Fidelius spell, the Death Eaters who gathered outside of 12 Grimmauld Place only knew that someone spelled Voldemort's name, but they could the specific location didn't find or suspect that was Harry Potter and his company. And remember: most of them were kidnappers, not Death Eaters. They didn't answer Voldemort directly, so there was no way they could have known that the Order's headquarters were around this place. I can assume that Voldemort would not inform all of his followers about the respective location, let alone the fact that Snape should be the one to reveal the location to others.

Of course, this doesn't explain why Voldemort himself stopped visiting 12 GP to see if Harry Potter was hiding there, especially after his failed attempt to catch him at Bill and Fleur's wedding. But considering that Snape was working for Harry and the trio had the portrait of Phineas Nigellus with them at all times (so Snape could know what they were up to), the former could be providing Voldemort with false information about the temporary visit to the place and nobody Find. And given that Snape was the most trustworthy of the Death Eaters, Voldemort wasn't going to question him any further.

TO EDIT:

Also, don't overlook the fact that Snape may just have told the secret to Voldemort himself. Voldemort would not have told Bellatrix or anyone else about the place, or Snape would have had it revealed to them because it makes no sense to do that. He wasn't a chip chat guy, and he wasn't a reason for her to know. It was the abandoned headquarters of the OoP, nothing more. With only Voldemort and Snape knowing the secret, I can't imagine the Dark Lord patrolling at 12 GP ...

As for the "plot hole" about Bellatrix who knows the location of 12 GP before the Fidelius spell; she may have known that, yes, but the Fidelius Charm, when placed on a building, reveals the building not even . It has nothing to do with address or location (although the Prior could be used by the Keeper to include a person in the inner circle of those who share the secret). Bellatrix could literally visit the street between 11-13 GP knowing that the headquarters is at 12 GP and doesn't see the building at all. Likewise, she could have told Voldemort of the address, or written it, or screamed, but neither Voldemort nor anyone else could see 12 GP when they were at the door. I imagine something similar would have happened if she tried to Apparate there.

Pryftan

"Perhaps we will know through Mundugus that he used the Imperius Curse on him." He confused him. He even reveals this to Harry in his stories at the end. But as for the Death Eaters (pretty sure it wasn't just kidnappers) around Grimmauld Place, that's a theory - they at least used the name Voldemort while inside. But Lupine implied that they were all over, too. It's certainly a plausible theory, but it's likely they would have been there anyway. I don't think there's a plot hole with Bellatrix at all, and that's not even controversial (well, it's still invalid, though).