Think of it this way - if you set your goal as beating the fastest record for the 100-meter dash, the logical conclusion is that you want to run the race as close to zero seconds as possible. The "breakdown of the tonal system" is the "logical extension" of the trajectory of music in the 19th century, whose goal had been taking harmony to the most emancipated place possible. Schoenberg himself talks about this and certainly views the atonal system in these very terms.
But there is always a reaction away from any dominant movement. (No puns intended.) Practically all the music that is not in the German tradition of increasing chromaticism can be categorized as a vague return to modality. Kostka and Payne even note this. Think for a moment.
The French school was dominated by Debussy and Ravel in the turn of the century and the first half of it. They worked, primarily, in the whole tone scale and the Dorian mode. (Check out Jeau D'eau, for instance.) Olivier Messiaen was also a central figure and not entirely atonal. He worked in "modes of limited transposition". Even in a piece like La Transfiguration, he wanders far, but never entirely away, from E major throughout the entire work.
The English school was dominated by figures like Vaughn Williams, Holst, Walton, and Britten. Look how much of their music employs the Church modes. (The Fantasia on a Theme by Thomas Tallis comes to mind immediately, but so does Jupiter, Somerset Rhapsody, any of the choral music of Walton and Britten and showing up prominently elsewhere.)
The Russian school is enormous and odd. But look at Petrushka by Stravinsky, Scheharazade by Rimsky-Korsakov. When not gushing with Wagnerian chromaticism (like Rachminanov) or verging on atonal (like everyone else), even the Russians resort to static harmonies and "tonal" centers without dominant chords.
Modality permeates music that is not along this central trajectory (as at least Schoenberg thought it) of Classical music. It seems that Classical music is obsessed with looking back at two things - primarily, the Classical age of drama (like the Monodists and Wagner) and, secondarily, the folk and Church music that provides a strangely removed flavor.
A gorgeous, concrete example of all this is the Somerset Rhapsody by Gustav Holst (previously mentioned). Check it out - he stays almost entirely in modes (Dorian, Aeolian, Lydian) with a few splashes of harmonic color. It's peaceful and almost fairy-tale-esque in tone. Then, for whatever reason, Holst decides to go Wagnerian (listen to the chromatic scales in the violas and woodwinds) at 6:15 in the recording until it fades away about halfway through the seventh minute. A wonderful example of the conflict present in most 19th and 20th century composers between a look back at modality and a look forward toward harmonic emancipation.