Dumbledore is talking c**p. There are no preconditions for passing the sword between wizards.
It's arguable that need and valour are needed for one to receive the sword. However, the sword could only be presented in these circumstances by the Sorting Hat. In all the other occasions that the sword changed hands, need and valour didn't enter the equation at all.
Let's look at the only two instances in canon where the sword 'presented itself' to people. Firstly, Harry takes it from the Sorting Hat in the Chamber of Secrets.
"Help me!...help me!..." Harry thought, his eyes screwed tight under the hat. "Please help me!"
Something very hard and heavy thudded onto the top of Harry's head, almost knocking him out. Stars winking in front of his eyes, he grabbed the top of the Hat to pull it off and felt something long and hard beneath it.
A gleaming silver sword had appeared inside the Hat, its handle glittering with rubies the size of eggs.
(Chamber of Secrets, Chapter 17, The Heir of Slytherin).
Secondly, Neville takes it from the Sorting Hat at the Battle of Hogwarts.
In one swift, fluid motion Neville broke free of the Body-Bind Curse upon him; the flaming hat fell off him and he drew from its depths something silver, with a glittering, rubied handle-
(Deathly Hallows, Chapter 36, The Flaw in the Plan).
Were Harry and Neville both in conditions of need here? If the risk of being eaten by a wounded Basilisk or burned alive count as situations of need then yep. Did they express valour? Certainly. Neville had just given Voldemort the defiant "I'll never join you" speech whereas Harry had said that Dumbledore was a greater wizard than Voldemort. Rejecting the supervillain demands some courage. Harry also calls for help from the Sorting Hat; Neville may have done the same.
Now compare this to the other known instances of the sword changing hands in canon:
- Harry giving the sword to Dumbledore.
- Dumbledore (surreptitiously) passing the sword onto Snape.
- Snape giving the sword to Harry.
- Griphook taking the sword from Harry.
None of these instances display "need and valour". Indeed, with Griphook I'd more accurately describe his actions as displaying 'treachery and cowardice'. The reason why the recipients above don't need to display courage to get the sword is because they are being given it by another wizard, not presented with it by the Sorting Hat. Without the presence of the Sorting Hat, the sword could change hands just as any other magical item could.
In the specific case of Dumbledore which was asked for in the question, no, he didn't take the sword under conditions of need and valour. Harry seems to have just left the sword in his study in The Chamber of Secrets and Dumbledore took it upon himself to place it in his study.
"And did you kill a Basilisk with that sword in Dumbledore's office?" demanded Terry Boot.
(Order of the Phoenix, Chapter 16, In the Hog's Head).
As for Snape giving the sword to Harry, Snape obviously hid the sword in such a way that Harry would have to do something (kinda) heroic to retrieve it. This was quite unnecessary. He did it on the instruction of Dumbledore, as the quote in the question states. But Dumbledore was, intentionally or not, misleading Snape about the need for valour. It's quite logical to say that if Dumbledore didn't take the sword from Harry in courageous circumstances then Harry hardly needed to show any additional courage when taking it back. Indeed, Harry basically thought as much at the time.
Where 'chivalry' entered into this, he thought ruefully, he was not entirely sure, unless it counted as chivalrous that he was not calling for Hermione to do it in his stead.
(Deathly Hallows, Chapter 19, The Silver Doe).
So Harry's dive is unnecessary. Although he and Ron do show bravery in retrieving the sword, others (including Dumbledore) took possession of the sword without showing any bravery whatsoever.
Conclusion: no, Dumbledore didn't display need or valour. And he didn't need to.