harry potter - Why could no one guess that a Basilisk was the monster?

In the Chamber of Secrets, till the end, no one knows that the monster of Slytherin and the creature behind the attacks is a Basilisk.

Dumbledore, of all people, should have figured it out using the following clues

There are the following observations from which at least Dumbledore could figure out that the creature was a Basilisk:

1. 
   

   Slytherin (and Riddle, perhaps?) being fond of snakes, it would be hard to miss that the monster of Slytherin would be snake-like at least.

   
   


2. 
   
   

   There are presumably a limited number of magical creatures that can petrify their victims.

   
   


3. 
   

   The fact that roosters were being found dead in suspicious circumstances was a clue to the Basilisk.

   
   


4. 
   

   Plus, it wasn't a satisfactory explanation that an Acromantula killed Myrtle, as she was uninjured, but dead.

   
   

Surely Dumbledore could put two and two together and figure out what the monster was?

I believe the clues pretty much narrow it down to a Basilisk, correct me if I am wrong.

After all, even if it takes​ a genius to figure it out, Dumbledore was the right man for the job!

But he presumably didn't

I say this because:

1. 
   

   He made no attempt to distribute mirrors among the students as a safety measure.

   
   


2. 
   

   Nor did he station hundreds of roosters in the school to kill the basilisk.(Surely even Riddle would have a hard time killing so many roosters quickly to save his basilisk?) This would have been an effective measure, as roosters living as far away as near Hagrid's hut were threats to the Basilisk, so many roosters in the castle would probably get the job done.

   
   
   

Of course these methods are kind of extreme, but they are helpful in saving lives.

I can understand if no one else could figure out what Slytherin's monster was, but why couldn't Dumbledore do so?

After all, all it would take is some digging. Dumbledore, highly intelligent, would probably figure it out easily.

Can someone give me a reasonable explanation for this?

Answer

Dumbledore was stumped by victims being Petrified and not killed

The one distinguishing feature of a Basilisk is its killer eyes. Getting an eyeful from a Basilisk is as good as getting hit by Avada Kedavra. Unsuprisingly, Dumbledore was stumped when he realised his students were getting Petrified, and not killed; students getting killed would have been telltale signs of a Basilisk inhabitant.

“She has been Petrified,” said Dumbledore (“Ah! I thought so!” said Lockhart). “But how, I cannot say. . . ”

“Ask him!” shrieked Filch, turning his blotched and tearstained face to Harry. “No second year could have done this,” said Dumbledore firmly. “It would take Dark Magic of the most advanced —”

Perhaps, as the latter quote implies, Dumbledore rather believed it was Dark Magic.

The roosters' killer might not have been a Basilisk

There are plenty of ways to kill a rooster, let alone plenty of creatures who could've killed them. Hagrid himself says:

“All righ’, Harry?” he said, pulling up the balaclava so he could speak. “Why aren’t yeh in class?”

“Canceled,” said Harry, getting up. “What’re you doing in here?”

Hagrid held up the limp rooster.

“Second one killed this term,” he explained. “It’s either foxes or a Blood-Suckin’ Bugbear, an’ I need the headmaster’s permission ter put a charm around the hen coop.”

I like to think that he went up to Dumbledore saying:

"Professor Dumbledore, sir, the roosters in the hen coop are dyin' mysteriously. It could be either foxes or a Blood-Suckin’ Bugbear (no sir, not a Basilisk!), mind putting a charm around the hen coop?"

Thus Dumbledore might have viewed the roosters being killed as an isolated incident.

Harry kept his mouth shut

This question and its answer implies that Dumbledore didn't hear the Basilisk hissing, which was a very important piece of the puzzle that Hermione figures out only with Harry's help.

Dumbledore clearly asks Harry whether he knows anything relating to the attacks, and Harry blatantly denies Dumbledore information.

“I must ask you, Harry, whether there is anything you’d like to tell me,” he said gently. “Anything at all.”

Harry didn’t know what to say. He thought of Malfoy shouting, “You’ll be next, Mudbloods!” and of the Polyjuice Potion simmering away in Moaning Myrtle’s bathroom. Then he thought of the disembodied voice he had heard twice and remembered what Ron had said: “Hearing voices no one else can hear isn’t a good sign, even in the wizarding world.” He thought, too, about what everyone was saying about him, and his growing dread that he was somehow connected with Salazar Slytherin. . . .

“No,” said Harry. “There isn’t anything, Professor. . .”

Out-of-universe

1. 
   

   The most obvious reason: Dumbledore didn't find out about the Basilisk because Rowling didn't want him to.

   
   
   


2. 
   

   A more unlikely one, but still plausible: Dumbledore wanted Harry to find out about the Basilisk himself. Because of the events leading up in Chamber of Secrets, Harry had become quite uncertain on whether he should've been placed in Gryffindor instead of Slytherin. Only at the end of Chamber of Secrets where Dumbledore explains to Harry the significance of him pulling the Sword out of the Hat, Harry finally realises he belongs in Gryffindor. Of course, this reason would probably mean that Dumbledore was a horrible headmaster who sacrificed his students for Harry Potter's wellbeing, and therefore unlikely.