I have found, via Tom’s method, a novel argument for the Condorcet criterion, suggesting that Approval can be improved upon by first ruling out a Condorcet winner. If the Condorcet loser criterion is also valid, it implies the following system, using score ballots, would be better still:

Look for a Condorcet winner. If one exists, he wins.

Repeatedly eliminate the Condorcet loser until there is none.

Let each voter approve all remaining candidates he prefers to the average remaining candidate and disapprove all remaining candidates he prefers the average remaining candidate to. The candidate with the most net approvals wins.
If the Smith criterion is valid, Step 2 should be replaced with:
 Eliminate all candidates not in the Smith set.
But if the Smith criterion is valid, my instinct is that a stronger criterion is valid, one that’s satisfied only by voting systems that always elect a candidate from the Porch Set (?), constructed as follows:

Select an ordered pair of candidates (the set does not depend on which ordered pair you start with, but the quickest way to find it is probably to start with the candidate with the highest total score first and second highest total score second).

Let each voter approve all candidates he prefers to the first candidate and all candidates he prefers to the second candidate and considers equal to the first candidate. Let him disapprove all candidates he prefers the first candidate to and all candidates he prefers the second candidate to and considers equal to the first candidate.

If the candidates with the most and second most net approvals have already been the ordered pair, the Porch set is the set of candidates that have won at least once since that first instance of that ordered pair. If not, let the candidates with the most and second most net approvals be the new ordered pair and go back to Step 2.
The Porch set is a subset of the Smith set and more appropriate to approval voting, where, unlike in plurality, the first frontrunner’s probability of victory relative to the second’s matters. Porch/Approval would thus replace Step 2 of Condorcet/Approval or Smith/Approval with:
 Eliminate all candidates not in the Porch.