...but it can be done, yes? Our sax player finds a way. It may not always be exactly as originally written, for sure.
With a fretted instrument like guitar/bass, transposing is relatively easy, since patterns remain essentially the same but just move around the fretboard. And if you're talking about *down-tuning* to transpose, then it obviously requires no effort or thought whatsoever.
Keyed instruments, by contrast, require much more effort to transpose. Re-tuning is usually impractical (executing a pitch raise on a piano can take hours). Transposing on-the-fly requires good knowledge of chord and scale fingerings for lots of weird scales.
For instance, lots of guitar songs are in E minor, which *normally* only has one sharp for a piano, which makes it an easy key and one of the first few you really learn how to play with and improvise. But if everyone else is down-tuned a half step to concert Eb minor, then now the pianist has to play with 6 flats. Yuck! Unless, of course, it's an electric piano with a transpose function. Then you can make playing in Eb minor look like playing in A minor (C major).
Similarly, for a standard Bb-tuned tenor saxophone, playing a song in concert E-minor would *normally* transpose to F# minor (A major), which has 3 sharps and while not a beginner key, is at least something most blues saxophone players would have practiced. But with a band down-tuned a half step, the tenor sax now has to play in F minor (Ab major) with 4 flats, which is a much less common key I would argue.
Anyway, the point is, it's non-trivial and you're lucky if you have a sax player that can transpose as easily as a guitar or bass player.