"You only need three colors" is a bit of a cheat, because it doesn't really work out in reality. You can use three colors to get a good color gamut (as your screen is doing right now), but to represent close to every color we can see you would need to choose a red and blue close to the edge of what we can perceive, which would make it very dim. And because human vision is weird you would need some negative red as well, which doesn't really exist.

Printing instead uses colors that are in the range we can perceive well, and whenever you want a color that is beyond what a combination of the chosen CMYK tones can represent you just add more colors to widen your gamut. Also printed media arguably prints more information than just color (e.g. "metal" colors with different reflectivity, or "neon" colors that convert UV to visible light to appear unnaturally bright)