One of the descriptions of Mozart's music using the example of Symphony No. 40 was the balance of chromaticism and diatonism. His melodies in that piece were highly chromatic using notes not in the scale but the way he harmonise helps "regulate" the dissonance.
The most extreme example was the development section of 4th movement of Symphony No. 40 where he used 11 out of the 12 tone (the only tone missing was the tonic) and Bernstein described that even in that section he was able to balance it. Also this is often used as an example of the predecessor of 12 tone technique by Schoenberg.
The implication that I get out of this is that it doesn't really matter how chromatic the melody is as long as you balance the melody with appropriate harmonisation it will still sounds coherent and make sense to listen to.
So what is the next step to follow the footstep of Mozart and take this balance of chromaticism/diatonicism to the next level of development?
Synthesise diatonicism with 12-Tone Technique
Use the 12 tone row as a pure generator of melody and ignoring all the other process involve in 12-tone technique to ensure the music is atonal (so ignoring the Hexachordal combinatoriality which is series of rules where you harmonises the tone row to produce hexachords)
Treat the 12 tone row similar to a chorale melody to be harmonised with traditional tonality.
The composer melody has to use all 12 tone tones before repeating itself from it's principle melody. However the underlying harmony underlying follows traditional harmonic rules. Once the prime tone row is completed it would be followed by another tone row (which could be transform via transposition/retrograde/inversion) and then another tone row indefinitely until the end of the piece. Each tone row is harmonise using traditional tonality and hence they will always be a tonal centre in the piece.
I guess people asked why not just create a tonal piece of music instead of following these rigid rules.
The answer is that by following the rules the tone row is not used to create atonal music rather it is used to maximised chromaticism. By maximising chromaticism you'll have music that has the same beautiful ambiguities that is present in Mozart music. This may well produce interesting results but still be comprehensible by the population (unlike majority of serialism)
The result you will take Mozart's balance of chromaticism/diatonicism to the next logical step. You would also unify 12 tone technique with tonal music and bring things to full circle by having 12 tone technique advance tonal music. You can also connect the influence between Mozart and Schoenberg explicitly (Schoenberg insisted that he is a student of Mozart and he is only following the tradition of tonal music).
anyway as someone with zero training in music composition, I have no idea whether this suggestion is feasible, or even desirable. However this is the crazy idea that popped in my head :)