Well, Schoenberg's mature music is not tonal. So are you against these terms entirely, or do you have a different term you'd like to suggest?
A piece is atonal if it doesn't work.
Whatever, folks, love'm or loathe'em, it is pretty much uncontested that Schoenberg's pieces 'work.'
The term atonal is itself an extreme misnomer, the Latin 'a' + tonal literally means "music without tones."
Of course, there is no such thing as per the definition of Music. The relative academic meaning is music without a tonic (and that means tonic triad)
center or home base.
Ultimately, the term has become an automatic turn-off or turn-away for people who have never even sat down to give atonal music any real and unbiased listening attention, which is why, imo, it is best dropped from the vocabulary.; I do know that as terms, Modal, Tonal and Neotonal have seen next to no such misunderstanding and abuse ;-)
Think, when a piece is described as tonal -- and just how much that alone really tells you about the piece... which is actually nigh unto zip-nada... other than to let you know it is not modal or atonal, lol. I can not think of the legions of misuses of the term atonal I have read or heard (Prokofiev = 'atonal,' The double fugue from Stravinsky's Symphony of Psalms = 'atonal.'
are but a few.
Modal, Tonal, Neotonal and Atonal are but categorical terms for various general
harmonic usages, labels that will tell you next to nothing about a piece other than its most general harmonic usage. Outside of academia, I don't see much practical, or legitimate, need or use for the terms. The only thing any of those four terms as descriptors might tell you is that a piece is not one of the other three.
Better, imo, "more or less dissonant," in context with the era from whence the music comes. That is more accurate, requires more effort vs slapping Tonal or Atonal on as (too superficial and easy/glib) identifiers. Rameau and Bach, near to exact peers by date -- and of equal 'greatness' in that how high up Mount Parnassus are the composers game -- are both quite dissonant from that era. Mozart and Beethoven were both quite dissonant for their eras. Hell, even Musica Ficta in its time was considered wildly dissonant, and to unaccustomed contemporary ears it can sound pungently tangy on the dissonance front.
Consider some of the more acutely angular and dissonant serial ('atonal') music from the 1950's, and then think on the soaring late romantic lyricism of Berg's Violin Concerto,
or Schoenberg's late Piano Concerto,
veritably littered with his lingering sentimentalism for all things older from 'German music,' the interpolation of simpler old song and dance forms and other like variants within. There is also Dallapiccola, whose atonal music bears the sundry stamps of the typically Italianate; lyricism, emotive drama, etc. They're all 'atonal' by category, yet... which is why I consider modal, tonal, neotonal and atonal as being of very little real service to the general listener.
I offer you and anyone else this challenge:
Since the term Atonal is a dreadful misnomer, come up with another term for this music other than atonal. Serial is apposite, Second Viennese School, too, though less pithy, while 'neotonal,' though it applies, is a Johnny come later -- and already taken. Have at it, then ;-)