@toby-pereira you might find the paper interesting, if I recall it isn’t extremely long and you can get the gist without digging into the detailed examples.
I think unfortunately the system I proposed is only well-defined if there is at most one unique Hamiltonian cycle (up to cyclic permutation) in the tournament. This is probably often the case in practice but in general is simply false for more than 4 candidates, and resolving the degeneracies in a canonical way opens a can of worms. I also never finished the analysis of whether the induction for existence (almost always) of a unique residual winner actually works even in that case.
I think I abandoned it after failing to find a proof or counterexample for 4 candidates, hesitating to mess around with 5 candidates, and then realizing the degeneracy issue. I don’t remember if I found anything potentially interesting but can review my notes.
Also the only reason I wanted to settle the case of reversal ties was to extend the existence criterion which according to your notion was maybe at least half artificial.