A good question, and one that I've often thought through without a satisfactory conclusion in my mind.
I've done a fair bit of machinist work as a young tad, and I have machined a number of "precision" items where slots/keyways were cut in shafts that I had turned from bar stock. While not the simplest of tasks, it is quite possible (given the right equipment and skills) to cut things to thousandths of an inch precision, to drill accurate holes into the side of the same, and so forth. Doing the same with a piece of maple with the appropriate tooling would be easy enough. Tedious, requiring a lot of attention to detail, but easy enough.
Fabrication of the keywork would be of a much higher magnitude of difficulty, but not beyond the pale - others have been doing it for a long, long time.
But, getting all of those skills together in one place is another thing entirely. If it wasn't all that difficult, we'd be hearing about Heckel of Burlington, or wherever else fine bassoons would be made under such conditions. And, we know that they are not.
However...
Just like the good old days, when everything turned out east of Suez was automatically termed either native art or junk, there is a tendency to assign the label "crap" to Chinese musical instruments. (I say Chinese here, but just as appropriate would be Pakistani, Indian or Indonesian.) And, while that might be true now, it can (and has) changed in a hurry.
While we now look at Japanese products as high tech, precision, and as the end result of quality fabrication techniques, it was not always so. 'Twas a time when Japanese products were anything but "quality". Prewar Japanese Mauser patent rifles were poorly machined (even before we were bombing them back to the Stone Age), their motor vehicles were junk, and their aircraft fabrication was as sloppy as something done by a second grader.
Remember tin toys from Japan, immediately post-war? Same thing. Japanese cars from the 1950's and 1960's were also on the "tinny" side.
But then, something happened. In the space of perhaps ten years, Japanese adherence to the quality assurance teachings of Demming and others worked a minor revolution. Where the Japanese cars I rented in 1970 while in Japan on R & R and medical evacuation were pieces of junk, the first Japanese car that I bought in 1977 was tight, well built and worth every penny that I paid for it.
Yamaha pianos have always been quality items, at least the ones that I've seen. Their saxophones are also rock solid, if not everyone's cup of tea. Their electronics are first rate. Even so lowly of a product as the commercial flush toilet is, when produced by Toyo in Japan, flaw free over thousands of units.
And, the ways of the rest of Asia are following suit. Korea, once the equivalent of 1960's Japan, is now producing high quality (if perhaps not first quality) computers, automobiles and consumer goods. Ditto Taiwan/Nationalist China, and (to a certain extent) Real China and India. (There are some pretty robust agricultural machines being turned out on the Subcontinent.
If Japan can do it, so too (eventually) can China and Indonesia and (shudder) India. It may take more years than any of us here have left to us, but eventually it could happen.
When I was enjoying myself to no end, out in the boonies of the Republic of Vietnam, I spent quite a bit of time with the ARVN troops stationed outside of their major city of Pleiku. (My tank was the last one in the line of vehicles protecting Highway 19, and their armored cavalry regiment in the area had an outpost a couple of hundred meters down the highway.) While we Yankees mostly had modern equipment, the ARVN had a odd mix of modern and obsolete stuff, among which were a number of obsolete and obsolescent US small arms.
One of the weirdest things that I got to see (out of some pretty weird stuff) was a pair of Thompson submachine guns. Both were quite old, but a quality firearm well cared for can last for many, many years, just like a good horn. The bluing/browning was a bit worn here and there, but otherwise they were in very good shape.
The oldest of the two was a commercial model of the weapon dating back to the 1920's, this by the patent and manufacturer markings on the receiver. Big fins on the barrel, sculpted front hand grip, calibrated sights, butt stock fittings and all, it looked like it had just been taken from the hands of one of Big Al's torpedoes. (Alas, no drum magazine.)
The other was patterned after the first US military version of the weapon, introduced into production around 1942 or so. I say "patterned", because it was actually a Chinese copy, produced somewhere in the depths of Nationalist China at some time during the long, drawn out war against the Communists and the Japanese. Much more severe in appearance, this weapon was finished to military standards rather than to commercial ones.
However, in overall attention to detail, such as mill work around the edges of the receiver, fitting of the bolt and H piece within the receiver, the cleanup work within the receiver and all of that, it was a better piece of work. Only if you looked at it very closely did you see that much of the finish work on the Chinese copy was done with files and abrasives rather than with milling machines. (The twist in the rifling was also different, presumably with the American made one being correct and the Chinese one being accomplished with whatever equipment the Chinese found to hand.)
Both weapons functioned just fine, with the Chinese one being slightly more accurate when used as a carbine, probably due to the increase in the rifling twist. They both sounded the same too, having the characteristic "typewriter" sound caused by the pieces in the receiver rattling along as they moved. (You'd know the sound in an instant if you would ever heard it - they sound like no other automatic weapon on the planet, and very much like a manual, non-weaponry, Remington or Underwood.)
(And, neither was a pleasure to shoot. The "Tommy gun" uses a very heavy bullet (for a pistol round) which, even with the Cutts compensator at the front, the muzzle climbs so much after the first two rounds that it is virtually impossible to aim. Don't believe what you saw on The Untouchables...)
Now, if some poor peasant with a roll of tools and some bar stock steel can pull something like that off in 1943 era mainland China, I would submit that pulling off a decent saxophone or clarinet or bassoon there is only a matter of time. Gunsmithing and musical instrument manufacturing are similar in a number of ways, and there are an awful lot of Chinese there wanting to make money from something - why not a Zhang of Guandong bassoon?
Maybe not this decade, and perhaps not even the next. But, with the shining example of Yamaha to use as a standard of comparison, it will happen, sooner or later. European and American (and increasingly Japanese and Taiwanese) wage scales are pricing our goods out of the market, and the third world will fill in the vacuum, sure as a gun.