Beating everyone head to head
is beating the field. The reason Begich wasn't picked by RCV is because he didn't have enough first choice votes. The problem with your analysis is that you are placing more value on a person's first choice than is warranted. You are assuming people care so much about their first choice winning that their lower choices are essentially irrelevant as long as their first choice is in the race. However, this is not necessarily the case. For example, what if someone liked their first two choices equally (or perhaps close to equal), but was forced by the ballot to rank one ahead of the other? That is a very different situation than if the voters did in fact have a very strong preference for their favorite over their second favorite, which is the situation you are assuming. Neither of these situations can be assumed from the data presented on a ranked/ordinal ballot.
With a cardinal ballot such as score or STAR, your argument could potentially hold water because we can glean the nuanced preferences from people's ballots. eg if someone ranked Palin 5, Begich 4, and Peltota 0, that's a very different story from Palin 5, Begich 1, Peltota 0, even though both of those ballots would look identical if they were squished into a ranked ballot format. The second case would support your argument pretty solidly since the voter didn't like Begich much at all, just a smidgen better than Peltota.
For the opposite scenario, consider if this race had been done with a 5 point score/STAR ballot where 40% voters give Palin 5, Begich 4, and Peltota 0, 40% voters give Peltota 5, Begich 4, and Palin 0, and 20% voters give Begich 5. It would be inexcusable not to elect Begich. Even though Begich was not the first choice for 80% of the population, an election like that would indicate such extremely strong support for him that you'll probably be inclined to point out that such an election would never happen in reality and I'm giving a contrived example. Which you're right, I'm not saying this is a likely scenario, I'm simply trying to illustrate how a compromise candidate could in fact be very strong, but RCV will still eliminate them if they don't have enough first choice votes. Placing a premium on people's first choice votes as RCV does is both unwarranted given the data represented on the ballots as well as harmful due to the issues it causes with monotonicity, not electing the Condorcet winner, etc.
If we want to insist on ranked ballots, we should be using a Condorcet method to count them.