I'll flesh out my comment, because in hindsight I think it's actually a suitable answer.
In short, the answer depends on your conception of tonal music.
To someone in the Schenkerian tradition, tonal music is completely linear/horizontal. Even the basic motion of V--I is an outgrowth of melodic counterpoint: with scale-degrees 7 and 2 both converging to scale-degree 1, we're really left with harmonizing 2/7 with either 4 or 5. If we harmonize it with 4, that's a diminished triad, which we know won't fly; the only other option is scale-degree 5, which creates the major V we're accustomed to. From this viewpoint, the late 1700's chordal Roman-numeral music of which you speak is actually almost completely based in the horizontal tradition.
As such, from a Schenkerian standpoint, it's neo-Riemannian theory and the Tonnetz that are overly vertical and not at all horizontal, even if neo-Riemannian theory claims to be based in what we call "parsimonious voice leading."
With this in mind, it's tough to say how this would inform performance practice, because there are two competing opinions (at least) on this issue, and they're basically diametrically opposed to each other.
It seems to me that a performer must be aware of how they conceptualize tonal music. If they think that neo-Riemannian theory is truly the horizontal answer to the prior vertical understanding, then it seems to me they might want to emphasize these horizontal connections and long-term relationships. But if a performer views neo-Riemannian theory as an outright downfall from the horizontal understanding of earlier music, then the performer would want to emphasize and bring out different aspects of the music.