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Most other voting systems would actually have elected Begich.
He was the Condorcet winner; voters preferred him over Palin and voters preferred him over Peltola.
That's a stretch, and you're likely assuming that all Palin voters would vote for Begich. Again Peltola already had 48.8% in round 1 and wasn't the incumbent.
I also don't think weaker wide appeal (beyond majority) is the best way, as that seems like a potential race to a position-less (or simply inoffensive but ineffective) candidate. Though in this case it seems close, at least if it's red vs blue moderates.
Also sure, if Palin voters would've voted strategically their side might've won. I'm not sure if it's because they fell for the trappings of FPtP, because they were unwilling to vote for a moderate and thus bet on the wrong candidate etc. But voting for the non-incumbent as their first vote would've been safer as it'd allow them to still be a Palin voter if Begich lost in round 1 as he did. I don't think the situation is terrible, as under FPtP the only strategy would be for Begich voters to vote for Palin (full stop) which clearly they didn't even want to do as their second choice.
So they actually published the complete ballot counts for this election.
With the ballots as cast, in head to head elections, Begich beats Peltola 87264-79126, and Begich beats Palin 100311-63249. This is mostly due to Begich getting more second place votes than either Peltola or Palin. For example, 33761 Palin voters put Begich second, 3437 put Peltola second, and 21526 bullet voted (i.e. put down no second choice). Similarly, Begich got about 10x the number of second place votes from Peltola voters that Palin got.
Score Voting activist Warren Smith has a write-up on viXra: https://vixra.org/abs/2210.0103