I'm going to say that canon is inconsistent when it comes to memories. There are instances of memories being used as evidence in the Harry Potter series.
In Chamber of Secrets, the diary Horcrux is actually a preserved memory of sixteen-year-old Tom Riddle; Dumbledore uses the remains of the diary Horcrux to establish that Voldemort is active again and attempting a comeback.
"Dumbledore's been driven out of this castle by the mere memory of me," [Tom Riddle] hissed. (emphasis J.K. Rowling, Chamber of Secrets -- Chapter 17)
"A clever plan," said Dumbledore in a level voice, still staring Mr. Malfoy straight in the eye. "Because if Harry here--" Mr. Malfoy shot Harry a swift, short look "--and his friend Ron hadn't discovered this book, why -- Ginny Weasley might have taken all the blame. No one would ever have been able to prove she hadn't acted of her own free will" . . . "Very fortunate the diary was discovered and Riddle's memories wiped from it. Who knows what the consequences might have been otherwise. . . ." (Chamber of Secrets -- Chapter 18)
Dumbledore was prepared to use the diary with its memories preserved as evidence toward Ginny Weasley's innocence. However, Harry destroyed the diary with the basilisk fang, effectively wiping all memories from its pages.
Throughout Half-Blood Prince, Dumbledore uses memories in the Pensieve as evidence to show Harry Tom Riddle's/Voldemort's actions, motivations, crimes, and plans.
In Half-Blood Prince Dumbledore recounts to Harry how Morphin Gaunt was falsely convicted of the murders of Tom Riddle Sr and Tom Riddle Sr's parents in the Riddle House. When the Ministry came to question Morphin Gaunt, he confessed to the murders on the spot, therefore not requiring either Veritaserum or Legilimency (magic that extracts memories). Dumbledore was able to procure a visit with Morphin Gaunt at Azkaban in the weeks before Morphin died; Half-Blood Prince says:
"But he had this real memory in him all the time? [Harry]
Yes, but it took a great deal of skilled Legilimency to coax it out of him," said Dumbledore, "and why should anyone delve further into Morphin's mind when he had already confessed to the crime? However, I was able to secure a visit to Morphin in the last weeks of his life, by which time I was attempting to discover as much as I could about Voldemort's past. I extracted this memory with difficulty. When I saw what it contained, I attempted to use it to secure Morfin's release from Azkaban. Before the Ministry reached their decision, however, Morphin had died." (Half-Blood Prince -- Chapter 17)
This is an example of a memory being used as evidence by the Ministry of Magic; although they had not reached a decision, they were considering it. It also implies that the Ministry of Magic uses both Legilimency and Veritaserum (another way of gaining memories, via potion) to collect evidence to be used in criminal proceedings.
In Order of the Phoenix, Snape says,
[Legilimency] is the ability to extract feelings and memories from another person's mind--" He then explains, "The mind is not a book to be opened at will and examined at leisure. Thoughts are not etched on the inside of skulls, to be perused by any invader. The mind is a complex and many-layered thing, Potter . . . or at least most minds are . . . " He smirked. "It is true that those who have mastered Legilimency are able, under certain conditions, to delve into the minds of their victims and to interpret their findings correctly. (Order of the Phoenix -- Chapter 24)
In Goblet of Fire, Dumbledore says,
"The Longbottoms were very popular. The attacks on them came after Voldemort's fall from power, just when everyone thought they were safe. Those attacks caused a wave of fury such as I have never known. The Ministry was under great pressure to catch those who had done it. Unfortunately, the Longbottoms' evidence was -- given their condition -- none too reliable." (Goblet of Fire -- Chapter 30)
Yet, Barty Crouch Jr. was convicted of the torture of the Longbottoms and sentenced to life in Azkaban, on what appears to be testimony from the Longbottoms themselves, which would be their recounting of their torture presumably from memory.
Regarding the original question, as others have mentioned, Lucius Malfoy was not witnessed putting the diary into Ginny Weasley's cauldron, so there would be no memory of that incident by anyone other than Lucius himself. Perhaps Lucius's own memory, if it had been secured, could have been used as evidence against him at a criminal trial. And, yes, Harry could have quite possibly proved Sirius's innocence. Regarding the return of Voldemort, Cornelius Fudge was determined not to believe Harry, despite reasonable and overwhelming evidence that Voldemort was back. I believe he would have rejected a retrieved memory from Harry if one had been obtained. It wasn't until the end of book five that Fudge accepts Voldemort's return. Harry was not the only one to see Voldemort at the Ministry of Magic following the battle at the Department of Mysteries in Order of the Phoenix. Dumbledore dueled Voldemort and was aware of his return. Furthermore, Voldemort was witnessed at the Ministry in the atrium by Cornelius Fudge and an Auror named Williamson. As well, Order of the Phoenix indicates that there was a large group of people in the atrium at the time Voldemort was sighted.