I think Roetz 4.0 has done some experiments with this iirc. It worked fine, but it was more like one of those multi-color filaments as there is no way to control the color coming out of the nozzle. He tried with 2 different colors I think, but his setup had 4 wires of filament being pushed through.

There is actually a relative long time between filament being pushed into the nozzle and coming out at the bottom. In between is a reservoir of molted and partially molted plastic, which is essentially a chaotic system. When extruding or retracting the filament, this is actually controlling the pressure inside that reservoir. This makes it extremely hard to get any kind of controlled mix.

You also have the issue of the number of filaments required. Remember CMYK only works when printing on white paper, the white is an important component to the end result. And those don’t actually mix those inkts. They just print little dots close enough together for our eyes to lack sufficient resolution and thus see them as one mixed color. For a lighter result less dots are printed, for a darker result more dots are printed and black dots added to go even darker.

So you would need to have 5 filaments being pushed into a single reservoir, which is a challenge. Then you’d never get perfect mixing, even if you machine the combination inside perfectly (look at how nice the machining on Roetz his nozzle was). The pressure inside would vary leading to some filament being pushed less that others, it’s a very unstable situation which makes it very hard to perfectly predict or control. And being it’s mixing inside that reservoir, there is very little control of how the color comes out. It wouldn’t be the same color from all angles due to imperfect mixing and the changing of colors would be a slower gradual thing, not instant. And the way the color presents is also dependent on the direction the nozzle is moving.

The amount of material being pushed in would vary a lot, where we’d want mostly the same continuous pressure on the nozzle side. And when moving reduce that pressure to prevent stringing. Controlling this with 5 filaments would be borderline impossible and would require a high flow large diameter nozzle. So only very large high speed printers would work. Or the filament would need to be thinner, so less material is pushed in, but then you run into compression strength issues in the filament.

I think for practical results, people either just paint it after the fact. Use a multicolor filament for cool colored results. Or use a different technique that offers prints in precise colors.

There is this super cool 3D printing technique from HP that is abandoned now, but does kinda work like you say with CYMK and a white material. It uses inktjet heads to eject a ceramic powder to 3D print. It can make high resolution 3D prints in full color and when baked the ceramic is very hard, so the objects can be practical to use. The big issue with this technology is the powder clogs up the nozzles on the inktjet heads. Just like with printers when the ink dries and the head clogs, it’s a nightmare. So those machines need constant cleaning and supervision. When they work they produce amazing results, but the technology is such a pain in the ass, they stopped making the machines. I think they still make the material and heads tho, for people who are already using the machines. I think they are often used in medical settings where they paid a lot for the machine and keep using them.