I'm no mathematician, but Arrow himself said he was speaking of ordinal methods: “And in my theorem I’m assuming that the information is a ranking. Each voter can say of any two candidates, I prefer this one to this one.”[1] That's not to say that cardinal methods don't have failure modes, but the particular set of interdependent failures and how pathologically one or more failure of them can appear is not described generally by Arrow's theorem.

How can a ballot capturing cardinal values be reduced to ordinal ones? I don't understand what you're saying here. In an election using a cardinal method, the slate of candidates can be ordered ultimately when summing results, but that's not the same information as the collective mass of ballot data.

1. https://electionscience.org/commentary-analysis/voting-theor...