If you really want simplicity though, I suppose you could use IRV as your tie breaker amongst the Schwartz set.
Which isn't to say it isn't a good system; but its definitely a draw back. Supporting IRV seems like a good first step in which we don't let perfect be the enemy of good.
Yes, graph theory comes into play for tie breaking (assuming we can agree that cyclic Condorcet paradoxes are ties of sorts), but the core of the idea is already understood by an overwhelming majority in concept if not by name.
Of course, if you assume consistent relative performance, then any single winner tournament that you construct among the candidates in a Condorcet election would choose the Condorcet winner of one exists, and some member of the Smith set in any case (though different initial bracket structure can choose different Smith set members in the absence of a Condorcet winner.)
A method which is summable for k=1 is good for this, one which is summable for some k>1 is less good, and a nonsummable method (like IRV) is, to me, unacceptable. [0]