What's the rationale for giving that group disproportionate power in choosing the winning candidate?
There is no rationale, because RCV was never designed to give voters equal power.
Ranked Choice Voting (also known as Instant-Runoff Voting, the Alternative Vote, or preferential voting) is essentially a mistake: a misapplication of the Single Transferable Vote system.
Single Transferable Vote is a multi-winner voting system, designed in the 1850s to elect entire slates of representatives at once. It is primarily associated with Thomas Hare, whose intention was to make parliament more representative of minority viewpoints:
Every detail of this scheme converges to one central point,—that of making the exercise of the suffrage a step in the elevation of the individual character, whether it be found in the majority or the minority. I disclaim for it, therefore, the title of a representation of minorities.
The idea is that after one candidate has been elected, the voters who supported that candidate with their first preferences are now well-represented in parliament, and their further preferences should no longer have weight. The desires of other, unrepresented voters are now given more weight, so that they too have a good representative, and after repeating this process to fill every seat, parliament has good representation of a variety of viewpoints.
The first candidate elected is not intended to be a representative of the entire electorate, but only of the largest faction, and the representativeness of the body is achieved by electing further candidates of different ideologies.
STV does not lead to strict proportional representation of parties, in the way that systems like party list do, but it typically produces a closer approximation than FPTP.
RCV/IRV was then "invented", possibly mistakenly, in 1871 by William Ware, who was introducing Hare's method to Harvard. He describes Hare's method in detail, correctly explaining how it should be used for multi-winner elections, but, while listing the advantages of STV, he says:
It is equally efficient whether one candidate is to be chosen, or a dozen.
It's not clear whether he fully understood the implications of using a multi-winner system to elect a single candidate, but this is cited as the first description of what we now call IRV or RCV.
So RCV applies exactly the same process as STV, but stops after a single candidate has been elected, meaning that they are not a good representative of the electorate as a whole, but only of a large faction.
Other voting systems were designed from the beginning with single-winner elections in mind, and do a better job of representing the electorate as a whole.
For example, Condorcet's "ranked choice" system considers all voters' preferences simultaneously when deciding the winner, not discarding some and keeping others like RCV does. It is able to elect a candidate who is preferred by the voters over all other candidates, even if not the first choice of many. Other "Condorcet-compliant systems" try to achieve the same goal while reducing incentives for strategy, etc.
Likewise, the more recently-invented cardinal voting systems like STAR, Score, or Approval voting try to find the candidate who has the highest overall utility, meaning they are the best representative of the electorate overall, or would make the voters happiest as a whole.