Dumbledore hired Trelawney to prevent Voldemort capturing her and torturing details of the prophecy out of her. Evidently, this was an unnecessary precaution, as when Voldemort took over he ignored her and let her carry on teaching.

Voldemort does seem to think he's got the gist of the prophecy by DH, but I can't imagine he wouldn't want to know the fine details of it. It might've contained explicit instructions on how Harry should go about killing him for all he knew.

I'm sure Snape would have done everything he could to shield Trelawney but if Voldemort had been at all insistent, I think it would have been beyond Snape's abilities.

On a semi-related note was Firenze still allowed to teach divination during DH? Harry saw him injured in the Great Hall so he must have been fighting from the beginning. It surprises me that the Death Eaters would have been happy to let a centaur teach their children.

 

The Death Eaters allowed Filch to remain and he was a Squib, so I don't see them caring overly much. Especially as Firenze was betraying centaur kind by teaching their own secret magic to wizards, thats like a goblin sharing the secret of their smithing at no real cost, I'd think the Death Eaters are pragmatic enough to take such a unique chance, especially as we know they do not really have a problem working with "lesser creatures" such a giants and werewolves.

 

Voldemort thought he would be invincible with the Elder Wand, and so didn't think he had anything to fear from Harry once he had that in hand. Otherwise, the only really sensitive information that might have been divulged in the Prophecy would involve his Horcruxes, but since Dumbledore didn't start targeting them immediately afterwards, he could probably conclude that those weren't compromised.

As for Firenze, I'm guessing very few of the Death Eater kids would have wound up taking Divination anyways, so maybe they just didn't care about him. They kept Professor Binns around even though everyone takes that class in the early years, he's objectively incompetent and they could have easily used his class as an outlet for more propaganda. They weren't the most diligent authoritarians ever.

 

We don't know that Trelawney had any memory of her prophecy, so Dumbledore may have just been trying to protect her more than the information she had. Even if he removes the memory or if she just forgets every true prophecy she gives(in book 3 it seems she forgets the second true prophecy), Voldemort may still capture and interrogate her. It's entirely possible, though, that he knows or suspects that seers forget true prophecies, or believes that Dumbledore would never let her keep the memory of something that important.

 

Also relevant: Snape was in charge of Hogwarts during the Dark Ministry, so it was up to him to make the most immediate changes to the school (and possibly protect allies who weren't actively acting against the Ministry). I have no doubt that the Dark Ministry would have updated the History curriculum within a few years, but Umbridge was busy prosecuting muggleborns, so that was just on the backburner.

As Trelawney had proven herself to actually speak prophecies, it was in Dumbledore's (and later Voldemort's) interests to keep her cloistered and relatively unharmed, in case Fate had something more to say.

 

Voldemort was busy looking for the Elder Wand for most of the year, he probably told Snape to go interrogate her instead of bothering himself with it. Then Snape would tell him that she forgets her own prophecies and that no amount of Legilimency could get it out of her, and Voldemort would believe him because he's convinced Snape's on his side.