Perhaps consider getting a C melody saxophone? It has a range in between the tenor and alto. It was used for a while in the jazz scene back in the 30/40s for the same reasons you list! I beleive some manufacturers still make them but you can also find some quality refurbished horns!

In more mainstream stuff, The Legend of Zelda: Tears of the Kingdom theme and the One Piece Live Action Soundtrack come to mind!

Transposing makes the fingerings the same throughout the family. Also they are actually perfect 5ths and 4ths apart. Bari (Eb), up a fifth to tenor (Bb), up a fourth to alto (Eb) up a fifth to soprano (Bb). Some of the fingerings are annoying for sure, but nothing compared to clarinet/bassoon lol!

And then saxophonists were like “the hell it does” *voicing ensues*