The tone quality of baritone and bass saxophones is, in my opinion, better suited to classical music than the alto or tenor.
I personally disagree 100% with that perspective, but there's another factor at work here. There is a reason that all the leading solo orchestral instruments are pitched in the same "high, but not shriekingly high" range: they project much better, while still retaining a rich, expressive tone quality. As lyrical and powerful a solo instrument as the cello is, the violin has multiple times as much top-tier repertoire. (Don't even ask about the double bass as a solo voice in classical music.) And what's true of the strings is doubly true of the winds, since strings tend to have greater ranges. Flute rather than alto flute; trumpet or horn rather than trombone, baritone, or tuba; Bb/A clarinet rather than bass clarinet or basset horn; oboe rather than English horn or bassoon.
The saxophone is a little unusual in that the alto is the leading classical solo voice by a wide margin, rather than the soprano. (Soprano repertoire has been increasing a lot in recent decades, however.) Maybe the French horn vs. trumpet situation is similar. However, the alto sax can easily reach into the range of the instruments pitched slightly above it, especially with altissimo. The bari sax, as nice as it may sound at times (and it does sound nice in this Hass concerto), is and will remain far, far behind. There are norms of expression in classical music that are just more readily achievable in the "standard" solo pitch range.