Most of the arguments Republicans made after that election were bad.
But that election was genuinely an example of a number of the unfortunate pathological edgecases in ranked choice.
In particular, it failed a number of the mathematical fairness criteria that people have come up with over the years to compare voting systems. Much of it stems from the failure to elect the Condorcet winner, Begich. Basically, Begich could have beaten either Peltola or Palin in a head-to-head election, but he had fewer first place votes than either so he was eliminated first and Peltola beat Palin in the last round.
So first, it failed 'favorite betrayal' - Palin voters would have been better off voting for Begich. It failed participation: if a bunch of Palin voters stayed home, Begich would have won and they'd be better off. It failed monotonicity: Palin voters could have defeated Peltola by voting for her. Obviously, it failed independence of irrelevant alternatives; Palin acted as a spoiler candidate to Begich.
All of which isn't an argument for regular party primaries + plurality, which is theoretically much worse. But it's the example advocates of alternative systems like approval or STAR will reach for for a while, just like Burlington used to be.