Ah, what I know as "figure-skating voting". I don't know very much about these except that Arrow's theorem can't apply directly to them because they have a different ballot structure; however given that Borda is something of a special case of score voting (in the same way as in crypto; random permutations are special cases of random functions meeting different needs), presumably there are some regimes where you can push it so that it has similar behavior.
Borda could be thought of as a ballot-restricted version of score voting, but so could plurality; you can’t expect ballot-restricted versions of reasonable systems to remain reasonable. Generally, there’s no reason for voters to cast Borda-type ballots unless you force them to.