When Albus Dumbledore told Harry that he and Grindelwald were both interested in the deathly hallows, he also told him but for different reasons.
As he then told Harry that Grindelwald was more interested in the elder wand and that he (Dumbledore) was interested in the resurrection stone and the invisibility cloak.
The elder wand was also called the death stick because of its history. As some people like Lord Voldemort believed that a person who wanted it had to kill the person who has it or die trying to become the master of the elder wand
Though when Grindelwald worked out that a wandmaker called Gregorovitch had the elder wand in his possession.
Grindelwald stole the elder wand but did not kill Gregorovitch.
As perhaps why Grindelwald didn’t is because Albus Dumbledore told him that he didn’t think it was necessary.
Though Voldemort did as he made horcruxes and he didn’t want anyone to know that, and he killed Gregorovitch after Voldemort had questioned him about the elder wand to find out who had it.
Voldemort also is known to work by himself, as he did when seeking for information about the elder wand.
I think also a lot of Albus Dumbledore’s past was kept a secret and that Albus Dumbledore’s Mother taught him and perhaps his brother how to do this.
As, they helped their Mum keep what happen to her daughter and their sister a secret. Also, Albus wanted to destroy the elder wand’s power as he didn’t die defeated.
So, if Draco hadn’t been successful at disarming his Headmaster, then Albus Dumbledore believed that Lord Voldemort couldn’t use the elder as Dumbledore wasn’t defeated even though Severus Snape did the Avada Kadava.
However, not all secrets stay that way. As Harry had to find out through Rita Skeeta’s book as her book made Harry realize there was a lot he didn’t know about the headmaster.
Harry also worked out that when Albus Dumbledore drank the potion in the cave which must have been the draught of living death that Albus Dumbledore had to relive Grindelwald doing the same unforgivable curse, on Aberforth and Arabelle that was done to Neville’s parents.
It was also unclear what happened.
Though Arabelle Dumbledore ended up dead and they were all unsure who did it but then Grindelwald decided to disappear.
Albus Dumbledore had also made an unbreakable vow to Grindelwald. Though he knew that Grindelwald was interested finding the elder wand.
Albus Dumbledore later on had admitted to Harry that why he was interested in the cloak because he thought at the time that the cloak would hide his sister. He later told Harry that because he (Dumbledore) had selfish motives at the time the cloak would not have worked for him and that he didn’t really need a cloak to hid.
Then perhaps why Albus Dumbledore had put on the ring is because he was interested in finding it, but the resurrection stone had been turned into a horcrux.
Albus Dumbledore also knew that he no longer needed it for himself. Then as the years went by Albus Dumbledore might’ve worked out his and Grindelwald’s mistake and knew that Voldemort would go after the elder wand.
He also made sure that Professor Snape told Lord Voldemort that he (Lord Voldemort) had to kill Harry himself.
This is probably why Albus Dumbledore looked momentarily pleased when Harry told him that Voldemort specifically wanted Harry’s blood.
As perhaps Albus Dumbledore knew or guessed right that if Lord Voldemort tried the avada kadava curse on Harry that the spell would not have worked. This theory also proved to be true when Voldemort tried to do the avada kadava using Lucius Malfoy’s wand.
Then what Albus Dumbledore might’ve also realized was that a person had to desire to possess all three hallows as he had worked out how Harry could survive again.
As why Albus Dumbledore came up with ways to help Harry instead of just going along with the one idea is because he didn’t really know which idea would work.
Also, maybe the three hallows which were the elder wand, the invisibility cloak, and the resurrection stone, worked in a similar way that finding the philosopher’s stone worked.
As that was perhaps why Albus Dumbledore knew that Harry Potter was a better person. Also, near the end of the story when Harry was talking to Ron and Hermione. Harry broke the elder wand in front of them. Harry chose to do this because he didn’t think it was right that anyone should have so much power.
He also did this after he had used the elder wand to mend his original wand.