In Goblet of Fire, after the last task why didn't they simply use the pensieve to restore Harry's memory instead of making him relive the events?
Wouldn't this provide valuable evidence to make everyone believe Voldemort is back? It would be easier to make a ‘fellowship' if everyone was aware of what was coming which was clearly what Dumbledore wanted.
So why didn't he use the Pensieve?
There are two aspects to this: one practical, one political.
Memories are not a foolproof record of events.
Skilled Legilimens can conceal their memories against “mind reading”, so Occlumency is out. Also, you can tamper with memories – witness Slughorn in Half-Blood Prince – and somebody of Dumbledore’s skill must be able to produce a convincing forgery.
That means that any memory provided by Dumbledore and/or Harry would be treated with scepticism.
And just because Harry remembers something doesn’t mean that it happened. Memory modification (whether by magic, hallucination or trauma) is well-established in the Potterverse. Given the proximity of these memory’s to the trauma of Cedric’s death, plenty of people would be sceptical of their accuracy.
Nobody who matters would listen.
The main person denying Voldemort’s return is Cornelius Fudge.
He is well aware of Dumbledore’s skill, and – as we learn in Order of the Phoenix – is paranoid about Dumbledore’s ambition. He is determined to ignore any evidence of Voldemort’s return, especially if it comes from Harry or Dumbledore. A memory won't sway him.
Most other Ministry officials are going to tow Fudge’s line. Even if you could convince them to witness the memory, I don’t think they’d be much help in convincing Fudge.
Using the Pensieve may have a been a plausible way to check details, it's not necessarily accurate, as if you remember from the Half-Blood Prince, the first memory retrieval of Slughorn's conversation with Riddle/Voldemort about dark magic and horcruxes you'll remember that it looks like it's been tampered with, or at least been edited in some way.
