The highly interesting thing is, brown absorbs all other colors but reflects red and green.
This may be a source of your misunderstanding.
The way humans perceive color is by having three types of photoreceptors in their eyes. Each of them has a broadish response function which means, that the "green" receptors react well to pure green light (as in 555nm), but also are somewhat excited by yellow (589nm) and hardly respond to red (>600nm). The same reasoning goes to the "red" receptors, which respond well to red (632nm, for example), but the yellow (589nm) also affects them somewhat.
Therefore, when you shine a red and green light onto a white screen, it reflects both red and green photons, exciting both "red" and "green" photoreceptors in your eye, which the brain interprets then as yellow. But similarly, you can simply illuminate the screen with a yellow light and get a 589nm wave, to which both mentioned types of receptors are sensitive. As a result, they're both excited and the brain again interprets (this time correctly) this as yellow. (see first figure)
Now the way pigments mix is slightly different. A pure pigment absorbs all light, except light of its color. Again, not perfectly, but with a relatively broad response, i.e. a green pigment may reflect some blues and yellows, but the main part is centered on the green. Similarly, a red pigment absorbs all, but red. So when you mix the pigments, qualitatively you're summing up their absorption functions. The presence of red prevents you from seeing much of the reflected green light, conversely, the presence of green prevents you from seeing much of the reflected red. However, in between, where the yellow lies, their response functions overlap and they reflect (although strongly suppressed), some yellow.
This dark yellow, smeared over a wide portion of the spectrum is perceived as brown. (see second figure)
Fig. 1
Brain identifies "red" and "green" receptors to both be strongly excited in roughly equal parts. It figures, the wavelength of incident light to lie at the point where they are equally sensitive and concludes, that incident light is yellow. However, it may just as well be equal amount of red and green photons.
Fig. 2
Red and green pigments have - qualitatively - these kind of reflection spectra. When you mix them together, you get the worst of both worlds: neither green, nor red is reflected particularly well, some residual reflection remains in between.
You are correct that LEDs just produce a mixture of 3 wavelengths that our eyes and brains combine into one "color." However, that doesn't make it less real in all cases. There are some "colors" like pink which don't exist in the rainbow because they are actually caused by a mix of two different frequencies of light being seen at the same time (a bluish frequency and a redish frequency). Most people consider pink to be a "real" color, so it would be reasonable by that logic to call the "aquamarine" produced by a LED to be just as real.
Of course, if you had a color like orange, which is part of the rainbow, you can see the differences between different ways to make us see that color. They make filters which only permit a very small window of wavelengths to pass. If you had one of these for an orange wavelength, a "pure" orange light source would shine through, but the LED would be blocked because it was making the orange color by a mix of red and green lights. While the human eye treats that the same as the single-wavelength orange, our narrowband filter can see through its trickery!
Of course, this can also work the other way. If I have a filter which is explicitly designed to filter out the green light of my laser at 532nm, and look at a laser beam (which is one frequency of light), I can't see anything at all. The filter blocked it all out. If I tune a LED package to output a "color" that corresponds to how 532nm looks to the human eye, it can be seen even through the filter because it's not actually emitting anything at 532nm. Instead, it is emitting a mixture of other colors which the human eye calls "green" when combined.
A very useful concept is the CIE colorspace which describes how humans perceive color very well. It demonstrates why a 3 color LED is sufficient to "trick" our eyes.
As for pulse width modulation, you are correct. While you can theoretically adjust the brightness of an LED by any number of methods, pulse width modulation proves to be cheap, easy, and works extremely well in most situations. However, you do have to make sure you switch often enough to keep the eye fooled into thinking the light is constantly lit. It sounds like your particular LED is a bit too slow to maintain that illusion.
Best Answer
This has as much to do with biology as with physics. The long answer on the biology is here. The biology in summary: the human eye has only three different "color-sensitive" elements, and uses a complex combination of the amount of response it sees from each of these to assign a "color" to the image.
Because there are only three sensitivities in the human eye, there are a variety of different techniques using only three basic (in some schemes called primary) colors to represent colors to humans.
The actual frequency of the light emitted from the part of the rainbow we call green may have the same effect on the human eye as something we get by mixing our blue crayon with our yellow crayon on a piece of paper. But it is easy to build a detector which will trivially differentiate between monochromatic green from one slice of a rainbow, and a mixture of the light reflected from blue and yellow crayon pigments.
In principle, humans might have evolved a different eye with four or five different "color" detectors, in which case the schemes needed to make color images would probably need to have four or five basic colors, and the images we see from our current three color representations would seem to be washed out, missing something important. But the eye didn't develop that way.