Saxaphone is 'reviled' in classical music because its tone is very crunchy and uneven compared to most of the other instruments of the band. I think in tone it is pretty similar to the bassoon, but even a chunky, honky instrument like the bassoon is very smooth compared to sax. Bassoon 'sings'[1], saxaphone 'yells'[2]. Given that, sax has very little place in a classical setting because it doesn't have that bell-tone or sine-wave sound that most other classical instruments try to achieve.

[1] https://youtu.be/cKBrnjxlKgU?t=34

[2] https://youtu.be/pGaUlferotY?t=59

There's also some history to it. Saxophone is a relatively new invention, dating only to the mid-19th century, well after what people think of as "classical" was mostly done. By the time classical music could adapt, it had already moved in other directions. Saxophone was adopted readily by marching bands because it's portable and loud.

Historically, "jazz" started in the 1890s in New Orleans, when black performers were first allowed to play music in public. They played what was available to them - marching band instruments. Saxophones, brass, bass drums and snare drums. String bass and piano were added as the music moved indoors. Banjo dominated the string section until the electric guitar was invented, as guitars aren't loud enough to compete with all the horns and drums.

Unvarying unvarying belltone or sine wave sound would be more appropriate for something focused only on melody and not on tone, which is the opposite of OP's point. Sax tone has a lot of variability and control.

The melodic advantages of the saxophone are mechanical, not tonal. For example, to play a scale fragment on saxophone, one need only move fingers, while continuing to blow, creating a smooth, legato sound that can be done very fast. Contrast with guitar, where a note must be fretted and then picked, two separate motions that are difficult to coordinate. Smooth legato playing at high speed is extremely hard. Trumpets have a similar problem, tonguing notes to make changes.

And yes, saxophones have a great deal of tonal expressiveness available, especially once overblowing is brought into play. It's just not pretty tonal expressiveness, compared to other instruments.

You can still do legato on guitar: https://www.youtube.com/watch?v=bTI2s4svE2s&t=1m24s

Meh. Legato on guitar comes at a cost of expressiveness and note choice. When hammer-on and pull-off are your only ways to sound a note, you lose all the coloration available from dynamics, palm muting, distance from bridge, pick angle, and the million other little things guitarists do to make a note special. And note choice? Scales of any substance will force either difficult position shifts, or legato-breaking string switches.

It can be done, and I certainly do it. But it can't be all that's done, or your sound falls flat.

But it fits poorly into an orchestra. Your tone needs to mesh nicely. There is a reason why the well known classical sax pieces tend to use it as a solo voice (e.g., Pictures at an Exhibition).

It's my observations as a jazz musician and a tone nerd. The buzzy, shrill reed sound of the saxophone is difficult for the ear. Trumpets are much closer to the human voice. I've come to prefer saxophonists who revel in the ugliness of its tone, and push its other forms of expressiveness - not just melody, but the marvelous squawky tones only a saxophone can make. (That said, I'm a huge fan of modern master Kamasi Washington, who has a luscious tone.)

Look at it another way... would you rather listen to doves, or geese? Tonal beauty is objective.

A trumpet tone is famously close to a pure sine wave. Among all instruments, trumpet offers perhaps the least natural texture. I suspect all good trumpet players have worked at enriching their tone to avoid sounding synthetic, especially when it's most likely, as in classical music.

(Tries not be be snarky) You didn't respond to any of my points. Why don't you start playing trumpet, see for yourself how 'easy to get to sound nice' trumpet is. Everything you say seems wrong to me, except when you say what you like.

Like Dizzy Gillespie once said, "None of them blow easy". Yes, it's difficult to get good tone out of trumpets. Or violins, or guitars, or drums, or any other instrument. But assuming a skilled player, the tone of trumpet is more appealing to our ears than the tone of saxophones. It's ultimately easier to have nice trumpet tone than nice saxophone tone. Doves vs geese (or more correctly, elephants vs geese).

I've put in a couple of solid decades trying to improve my tone, and practice regularly to work on tonal issues. I appreciate the difficulty that any instrument represents.