The timeless popularity of this tune among jazz players is often remarked upon. Common explanations for this include the step-wise descent of parallel thirds between the bass and upper voice, mediated by a progression of chords through the cycle of descending fifths. Also, the tune notably features chord roots on each of the twelve chromatic tones. The resulting harmonic saturation provides ample fodder for improvisers.
I suspect that there is more to the story, however, as the kind of musical fecundity offered by "All the Things You Are" can never be so easily explained.
One question that strikes me as more difficult than it would appear at first is that of the tonic: what key is this tune in? The obvious answer is A-flat, the key in which the song concludes. The opening phrase tonicizes A-flat (as well as C), and the next phrase tonicizes the dominant, E-flat (as well as G). As Milton Babbitt suggests, the bridge tonicizes G and therefore acts as an "extended leading-tone" to the tonic A-flat. The non-tonic opening is described by analysts influenced by Schenker as an "auxiliary cadence," which means, as Michael Buchler explains, that "from a Schenkerian perspective, [the] refrain opening [does not] actively participate in its overall fundamental structure."
But does this interpretation underestimate the role of F minor?
F minor is metrically accented, appearing as it does in bars 1 and 25, and is prepared in both cases by its dominant. Also, the first eight-measure phrase could be heard as a rather typical prolongation of F minor through its relative major (A-flat) to its dominant (C). This i-III-V prolongation is then repeated at the fifth in the second phrase. An F-minor hearing is bolstered by the hyper-metrical "strong-beat" bass descent of F-E-flat-D-flat-C, supporting melodic tones A-flat-G-F-E.
If the bridge's G harmony represents an "extended leading-tone" to A-flat, could we not also say the same of the bridge's E harmony vis-a-vis F? If we posit an F-minor tonic, the G and E harmonies of the bridge could suggest upper and lower neighbor tones.
Finally, the D-flat7#9-C7#9 riff used by performers like Charlie Parker as both "intro" and "outro" points strongly to F minor as a gravitational locus.
A-flat undoubtedly has a strong claim to the tune's tonic, as the last sixteen bars clearly outline vi-(I)-IV-ii-V-I in A-flat. But the first twenty-four bars seem to make more sense as prolonging an F-minor tonic, where the E major at the end of the bridge behaves as a third-related substitute for the dominant: i-V-v-V/V-VII-(V). (E major and F minor are of course also related by SLIDE transformation since they share the same third.)
Could a case be made here for the "double-tonic complex" as described by scholars like Robert Bailey and Deborah Stein? Developed as a tool for analyzing nineteenth-century lieder and other music, the double-tonic complex is characterized by, among other things, a sense of oscillation between the two equally-weighted tonics. Jazz performances of "All the Things You Are," with their repeating improvised cycles through the tune's harmonic form, certainly seem to oscillate between F minor and A-flat.
This feeling of tonal ambiguity -- supported by both "auxiliary cadence" and "double-tonic complex" interpretations -- helps to illustrate the sense of the tune's lyrics. The lyrics are hopeful, longing, anticipatory, future-oriented. The tonic goal is continuously withheld, just as love is withheld in the song's lyrics.
It is important to stress that jazz performances of American Songbook pieces resist categorization along the standard teleological lines of Western music theory due to their cyclic and thus perpetually unfinished nature.