Why didn't Dumbledore verify through Legilimency whether Sirius was guilty of betrayal?

Question

Dumbledore was not only a close friend of the Potters; he was also the one who recommended the Fidelius as their best chance. It must have been a rude shock, when their best friend betrayed them. Dumbledore is also portrayed to be a skilled Legilimens, when he uses legilimency on Kreacher in Harry Potter and the Order of the Phoenix.

“And Kreacher told you all this… and laughed?” he croaked. “He did not wish to tell me,” said Dumbledore. “But I am a sufficiently accomplished Legilimens myself to know when I am being lied to and I - persuaded him - to tell me the full story, before I left for the Department of Mysteries.”

Dumbledore was also Chief Warlock of the Wizengamot. He had used legilimency on Morfin Gaunt, as he had access to Azkaban inmates, owing to his position in wizarding society.

“Yes, but it took a great deal of skilled Legilimency to coax it out of him,”

Why didn't he try the same on Sirius? Why didn't he use Legilimency on him to check whether he was actually guilty?

Answer 1

Sirius did not act like an innocent man

"A crater in the middle of the street, so deep it had cracked the sewer below. Bodies everywhere. Muggles screaming. And Black standing there laughing, with what was left of Pettigrew in front of him… a heap of bloodstained robes and a few — a few fragments —"

Laughing near a pile of corpses is not a good sign. It indicated that he had enjoyed killing them.

James Potter chose Sirius Black as Secret Keeper, but Dumbledore wasn't happy with that choice. It was Sirius' idea to secretly use Pettigrew instead.

"James Potter told Dumbledore that Black would die rather than tell where they were, that Black was planning to go into hiding himself… and yet, Dumbledore remained worried. I remember him offering to be the Potters' Secret-Keeper himself." "He was sure that somebody close to the Potters had been keeping You-Know-Who informed of their movements," said Professor McGonagall darkly. "Indeed, he had suspected for some time that someone on our side had turned traitor and was passing a lot of information to You-Know-Who."

Sirius' track record wasn't quite good

Dumbledore was very much justified in not trusting Sirius. Sirius tried to kill Snape at the Shrieking Shack by exposing him to a werewolf. In one stroke, he had endangered Snape's life, and Lupin's secret as well. He was indeed lucky to have escaped expulsion.

Therefore, Dumbledore had no reason to suspect Sirius' innocence at all, because he never trusted him in the first place. He performed legilimency on Morfin, because he was driven by the desire to know more about Voldemort's family.

Answer 2

Legilimency isn't foolproof. As a matter of fact, a good occlumens can hide what's on their mind. Sirius is a formidable wizard, legilimency would not have confirmed 100% since he could be guiding the information in his mind that a good legilimens may acquire.

Even if legilimency from Dumbledore confirms he is not guilty, it could still be false information. Snape used occlumency to successfully deceive Voldermort into thinking he, Snape, was loyal.

In Harry Potter and the Order of the Phoenix, Snape tells Harry about occlumency.

"Occlumency, Potter. The magical defense of the mind against external penetration. An obscure branch of magic, but a highly useful one."

Also Memory can be modified.

A quote by JKR from Bloomsbury Live Chat in 2007:

J.K. Rowling: They are two different charms. She has not wiped her parents’ memories (as she later does to Dolohov and Rowle); she has bewitched them to make them believe that they are different people.

Of course, Dumbledore may be able to work pass these defenses as he did with Morfin, however, the evidence brought forward against Siruis was quite substantial as stated by @Simpleton.