Her twinkle little star is entirely explained using typical, basic diatonic reharmonization techniques, like dominant chords and substitutes. And yes, it's all tonal in the key of Cmajor.
Start out on Cmaj. Move to Fmaj. Back to Cmaj. (I IV cadence)
Then move to Fm. A common trick because the #v (g#) has a dominant role, so you often see Fm, Dm7b5, Abmaj, Caug as part of a cadence in C, because each of those chords contain the g# and resolve back to (reinforce tonality of) Cmajor.
Back to Cmajor. Then Cdim (Idim). Again a common device in jazz and standards. This resolves to Gmaj (because the bottom two tones move down chromatically).
From Gmajor, she goes to Gm/Gm7. Three possible explanations, equally valid, seen many, many times in popular tunes:
(a) A typical key modulation, where the Cmajor chord becomes the dominant for the key of F, and hence the use of the IIm (Gm) as part of a IIm-V cadence in the new key.
(b) Bb major chord that is part of a modulation to the key of Ab (Ab has a dominant function in the key of C)
(c) key modulation from Cmajor down a whole step to Bbmajor. This sets the stage to come back chromatically to Cmajor as follows: Bbmajor > Bdim > Cmajor. (Bdim is a dominant chord in the key of C)
After Gm (Bbmaj), a dominant (G7 or Bdim).
And then back to C .... ? but wait, at this point, she does a fakeout. Instead of back to C, she chooses a random tone (C#) making it the only "atonal" tone in the song. And she does not harmonize the tone either, therefore truly making it random (and meaningless.)
Apart from this last tone, the rest of the song are cadences and substitutions that reinforce tonality in C major. Very typical chords you see over and over in many songs. Dating back decades if not hundreds of years. Just thumb through the Real Book.
If you harmonize the last tone as bII7alt (Db7alt) then you've got another typical dominant chord substitution in the key of C major.
Yes, she's using 12-tones, but it's all standard diatonic type of stuff. If you want a simple way to hear it, just play the block chords in the LH:
I IV I IV I IVm
I Idim Idim V
Vm V Vm V7
now if you eliminate the LH chords, and just play the melody, the tonality is still present, because the sequence of tones continues to establish the C major chord. The departures from the key (d#, f#, g#), are followed immediately by returns to harmonically neutral tones in the key (d, e, g) and the sequence in which they are played helps to "arpeggiate" the standard IV V I chord progression we are accustomed to hearing in C major.
Therefore, to answer the thread question, the supporting chords reinforce, but are not necessary for the tonality. Her melody by itself does it (the sequence of melody tones allows you to infer the underlying chord.)
It's not the "12-tone" music you normally associate with people like Shoenberg. And in my opinion rather misleading to imply that it's something "novel."