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Brass instrument 

A brass instrument is a musical instrument that produces sound by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips. Brass instruments are also called labrosones or labrophones, from Latin and Greek elements meaning 'lip' and 'sound'. There are several factors involved in producing different pitches on a brass instrument. Slides, valves, crooks (though they are rarely used today), or keys are used to change vibratory length of tubing, thus changing the available harmonic series, while the player's embouchure, lip tension and air flow serve to select the specific harmonic produced from the available series.

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all produce the same texture of sound

Do they? Some have conical bores (tubas and euphoniums) and produce a relatively warm sound. Some have cylindrical bores (trumpets and trombones) and produce a relatively bright sound. French horns are also quite warm, but that is because they play on higher harmonics and have a less direct sound than other brass instruments. While they may sound similar, there are noticeable differences between the timbres of various brass instruments. If a trumpet, a trombone, a french horn, and a tuba all played a middle c, these differences would be very apparent. >

I agree -- even within a traditional brass quintet, there are variations that can change up the sound. Bass trombone or baritone instead of a tuba, euphonium for a trombone, and piccolo trumpets or flugelhorns for a trumpet or cornet. 
I would say that the Toronto-based Canadian Brass quintet would provide a great counterexample for OP -- they've released several dozen albums in a variety of genres from Renaissance to ragtime. I wouldn't say it was easy, but it's clearly doable. Sure, some are original compositions but most are arrangements. >

The problems you've identified arranging for a brass ensemble would also apply to a string quartet (violins, a viola and a cello) which are  different size versions of the same thing. Same for a SATB vocal ensemble, which is made up of more or less identical singers.
What about writing piano music where you have 88 keys that are nearly identical in every way other than pitch?
I agree with you that arranging music in general is hard, but in my experience arranging music for similar sounding voices does not make it harder. >

Just because it's hard for you doesn't mean it's hard for someone else, just like something being easy for you doesn't mean it's easy for someone else. This applies even to things that seem similar, like one is easier than another. 
Look at running, for example. Sprinters may consider running a marathon difficult while marathon runners may consider it difficult to run over 20 mph. >

Are you familiar with New Orleans brass bands? Hot 8, dirty dozen, rebirth. It's one of my favorite genres of music. And many traditionally were not particularly highly trained musicians, just people working with what they had. >

Okay so I might not entirely change your view here but I can offer some perspective. I've gotten pretty deep into electronic music production so im coming at this from that point of view. Pretty much Sawtooth waves are the Brass of electronic music. They generally make up the big blasting chords and are the backbone of the tonal part of a piece (like excluding percussive stuff). Saws are used EVERYWHERE in electronic music and can be used in very low bass sounds and big block chords and very high leads. Similarly brass handles bass sounds very well in addition to being able to play high, solo melodies and chords. Like saws, they're kinda the jack of all trades. While they can do a LOT of things, they are kinda samey sounding at the end of the day and just having them blare in your ear can get very fatiguing. Just like how an electronic track wouldn't sound pleasant if it was all saws, a concert piece that is all brass would also be pretty unpleasant. So while they can do a lot, if it's just brass it won't work too well. But that doesn't change the fact that the brass are absolutely necessary when playing a piece that requires lots of grandeur. Of course there are pieces that can be played without brass and sound good but I'd argue they are limited to a more calm and tranquil mood (unless there is some other funky instrument to fill in for brass section)
sidenote kinda thing: brass sounds have especially high amounts of harmonics so they generally fill a lot of the frequency spectrum  (similar to a guitar in rock, or saw) Taking that into consideration, using all brass quickly fills the frequency spectrum and pretty much starts moving towards the concept of white noise which is why I think that honestly you're pretty much correct, it just gets too noisey. Some ppl like that tho idk >


So you just end up with a singular, constant texture of sound, whereas woodwinds can produce a more varied texture of sounds.

Let's accept for a moment that that's true, for the sake of argument. 
There's more to music than just timbre and texture.  Rhythm & harmony are also important.
Look at music for 4 handed piano or for organ.  Piano on 4 hands can play 20 simultaneous notes of an identical timbre.  Organs can play 12 notes.
How do you arrange for that?   You separate things into a few distinct components, like a few "voices" or lines playing chords or a melody. >

You say “on trombone you can change the length of tubing on demand”. In any brass instrument with valves, you can also change the length of tubing on demand. That includes trumpet and tuba. I would argue that drawing a distinction between tuba and trumpet vs trombone on the basis of ability to change the amount of tubing is incorrect. Instead it should at least be based on the range of lengths available to the instrument, size of the mouthpiece, and diameter of the tubing. >


Whereas woodwinds have ... saxophones

If you get to claim saxophones for woodwind, I get to claim didgeridoos and shofars for brass, which puts the lie to  

you just end up with a singular, constant texture of sound >

Saxophones are in the woodwind section. It's not really a claim >

And didgeridoos are in brass. Ball's in OP's court here. >

But they are not a brass instrument. A saxophone uses a reed, so it is a woodwind >


The view of most scholars (see organology) is that the term "brass instrument" should be defined by the way the sound is made, as above, and not by whether the instrument is actually made of brass. Thus one finds brass instruments made of wood, like the alphorn, the cornett, the serpent and the didgeridoo, while some woodwind instruments are made of brass, like the saxophone. *

Counter-intuitive, sure - but still.