Wondering who killed Ariana Dumbledore

This priori incantatem spell is still hanging around in the back of my head.

A useful little spell, isn't it? It causes a wand to spew forth, in reverse chronological order, all of the spells it has cast in the past. How far back it goes, I don't think is ever established, but it can be used in a variety of ways to understand how events transpired, to interact with people who died under that wand, to prove guilt or innocence even.

Useful, yes, but not used often enough, I think.

For example, the events of The Half-Blood Prince and The Deathly Hallows establish very clearly that Albus Dumbledore is haunted by the death of his sister, Ariana, and the thought, the fear that he himself could possibly have cast the spell that killed her.

His fear and remorse show themselves when he drinks the potion that protects the Slytherin's Locket in the middle of the lake in the cave by the seaside. He becomes emotional again in the "King's Cross" scene at the end of The Deathly Hallows, telling Harry that he fears that he himself cast the spell that killed Ariana.

Which begs the question: why, at some point in the 100 or so years between the Ariana's death and Dumbledore's own demise, did he not think simply to perform priori incantatem on his own wand and perhaps on Aberforth's as well. I will accept that Dumbledore would not have had access to Grindelwald's original wand but, if he tested both his own and his brother's, that should be sufficient to prove who cast the killing spell.

Okay, if priori incantatem is time-limited (or limited in the number of spells it can spew forth), perhaps Dumbledore could run out of time... but why wouldn't he think of it immediately after Ariana's death? He was a masterful wizard at the time. Perhaps he was too distraught and, by the time he thought of using this spell on his own wand, he had run out of time.

The situation begs another question: in all of Harry Potter lore, we have seen only one spell that kills its target (Avada Kedavra) -- does Dumbledore's reaction to his possible part in Ariana's death mean that he, Aberforth and Grindelwald were duelling to the death that fateful day?

Were all three of them throwing the Avada Kedavra around as they fought a boys' fight?

Or is is possible that an already weakened Ariana succumbed to a lesser spell?