Olaf's determination that he will find Anna and Elsa a tradition, and a great Christmas. He might be Comically Missing the Point in the execution, but his intentions are very loving.
The fact that the point of the short is that Olaf doesn't need to "find" a Christmas tradition from some other family, but that he himself is the Christmas tradition. Cue group hug between Elsa, Anna and him.
Said tradition being that Anna slipped a drawing or hand-made doll of Olaf every year under Elsa's door when they were separated from one another. Especially since Elsa seemed very happy to receive them and was always very eager to hear Anna's signature knock.
By the third flashback, Elsa is waiting at the door for Anna's gift.
Anna and Elsa immediately understanding Sven trying to tell Kristoff that Olaf is in danger, freaking out, ordering the bell to be rung, and rounding up a search posse to find their tiny snowman friend.
Elsa going door-to-door in person to find Olaf. Not only does it show how much she cares about Olaf, but it's heartwarming to see her interacting (even though that part doesn't have a voice track and only is shown) with the people of Arendelle. Judging from their reactions to her, she seems to be a well-loved queen, and the people don't find it even that weird/surprising that the Queen herself just goes door-to-door.
The fact that Anna and Elsa are basically having a conniption at him being missing.
Every Christmas, Anna gave Elsa an Olaf. This year, they both get an Olaf - the actual Olaf.
How everyone treats Olaf. Given that he's a Snowlem made by the Snow Queen, the villagers have good reason to fear him, but instead are fine with him now.
The whole town explaining their traditions to the beloved little royal Snowlem.
Then, when he's lost, everyone comes out in the middle of the night to look for him.
Even though Anna doesn't hear when Elsa comes to her bedroom to say she owes her an apology, she is still happy to see her in the attic, as if she doesn't need her apology to forgive her.
Besides the whole official Ice Master & Royal Deliverer job, Anna and Elsa show how much they value Kristoff's presence by appointing him and Sven with bringing the Yule Bell into the courtyard, and then, once the bell is hoisted into the alcoves by the villagers, dropping the rope down to the sisters for them to pull on.
One that exists thanks to Fridge Brilliance: The staff seem very much okay with Olaf surprising them and nearly giving them heart attacks, with their reactions suggesting they're quite used to it. Since Olaf is the embodiment of Anna and Elsa's childhood love and innocence, it's entirely possible that the sisters may have done this very same thing during the years before Anna got hurt, when they were both little, before anything bad happened, and when they were small enough to be able to hide just about anywhere and had endless excitement and joy. It was always fun to see people’s reactions to being surprised, especially when those people were used to seeing one of the princesses randomly jumping out from behind a curtain or bush and while it did startle them a little, they’d always overemphasize their reaction just to make it fun for them. It became a little game the staff could share with the two little girls they loved so much. But then all that had to stop when the gates were closed and the staff got cut, and the castle became quiet and empty. Anna still tried every now and then, but it wasn’t as fun without Elsa there with her. So eventually she stopped too. And the remaining staff didn’t know they’d miss something so simple until it was gone. Then comes living Olaf, and the snowman - with a little bit of Anna and a little bit of Elsa - quickly started to revive the surprise tradition, even before the holidays started. And for the staff who had stayed in the castle for all that time, they couldn’t be happier to play along again.
