Independence of Irrelevant Alternatives and the Condorcet Loser

https://electowiki.org/wiki/Condorcet_loser_criterion says:
Given a threecandidate Condorcet cycle, it's always possible to eliminate a candidate who didn't win so that the winner changes. Thus the Condorcet loser criterion is incompatible with independence of irrelevant alternatives.
Is this correct?
Consider two systems. System A is straight Score and System B is to eliminate the Condorcet loser (beatenbyallpairwise candidate) if there is one repeatedly and then apply Score to the rest.
If there are only three candidates in the election and they form a Condorcet cycle, there is no Condorcet loser so systems A and B produce the same outcome. How does this show a dependence on irrelevant alternatives? Isn't plain Score famously compliant with IAA?

@jackwaugh You missed an important part of the quote.
In an election with only two candidates, the Condorcet loser criterion implies the majority criterion. Given a threecandidate Condorcet cycle, it's always possible to eliminate a candidate who didn't win so that the winner changes. Thus the Condorcet loser criterion is incompatible with independence of irrelevant alternatives.
Say you've got an A>B>C>A cycle. Without loss of generality, A wins in some method that passes the Condorcet loser criterion.
Now remove B. C beats A headtohead so must now win (A is the Condorcet loser with only A and C). Given that removing B has changed the winner from A to C, the method must fail IIA.