All things considered, I think the events surrounding Anna’s freezing were especially nightmarish to Elsa - even more than we see when we first watch the scene.
Let’s outline it briefly. Hans tells her than Anna has died even though he’s lying. Elsa falls to her knees. Hans tries to kill her but is foiled by Anna, who - clearly not dead until that point - then proceeds to completely freeze right next to Elsa. She notices and starts sobbing.
Yes, she’s heartbroken. That much is evident. But more so than we initially realize.
You can see the grief, the resignation, the utter hopelessness radiating from her body as she falls to her knees. She believes her sister is dead, so she gives up on trying to get to her. She gives up on hoping for her safety.
But then Anna comes along and sacrifices herself for Elsa. Even though Elsa isn’t looking when it actually happens, she can hear Anna’s scream. She’s also not stupid, so she knows that the only way for a frozen Anna to have gotten in that particular spot is for her to have walked/ran there herself and then frozen just milliseconds before.
It takes a moment for her to realize it, but it finally clicks that Anna had just frozen. Just then. Not before, when Hans lied about her death.
She realizes that she gave up on her sister’s life while she was still alive.
This revelation comes after the fact, when there is absolutely nothing left she can do about it except embrace her statue of a sister, touching her in earnest for the first time in thirteen years.
Think about the guilt she must have been feeling as she stood before Anna: the guilt of giving up on her sister, the guilt of not even looking at her as she froze entirely, the guilt of realizing that she had effectively killed her last family member and her entire kingdom to boot.
Think about how she must have felt, reacting to the revelation of the same death twice in mereminutes.
And then imagine how indescribably immense her relief must have been when Anna was suddenly warm in her arms.
But even though Anna thawed, Elsa still has to carry those memories with her. We think at first that her sadness vanished with the ice, but she will always remember that she gave up on her sister’s life because she believed a lie.