I had experimented with some photo printing services and came across one professional level service that offered pigment inkjet printing (vs much more common dye inkjet printing). Their printers had 12 colors of ink vs the traditional 4. I did some test photos and visually they looked stunning.
Have you looked at the actual ink colors? Printing is a very different story. They’re not using 12 primaries, they’re using multiple gradations of the same primary. I don’t know which ink set you used, but 5 different grayscale values is common in a 12-ink set. Here’s an example of a 12 ink set:
https://www.amazon.com/Xeepton-Cartridge-Replacement-PFI4100...
There’s only 1 extra color there: red. There are multiple blacks, multiple cyans, multiple yellows, and multiple magentas. The reason printers use more than 3 inks is for better tone in gradations, better gloss and consistency. It’s not because there’s anything wrong with 3D color models. It’s because they’re a different medium than TVs. Note that most color printers take 3D color models as input, even when they use more than 3 inks.
You see this all the time with professional lighting fixtures as well!
For example, the ETC Source4 LED Lustr X8 has: Deep Red, Red, Amber, Lime, Green, Cyan, Blue, Indigo[0]
RGB LEDs are pretty crappy at rendering colours as they miss quite a lot of the colour spectrum, so the solution is just add more to fill in the gaps!
[0] https://www.etcconnect.com/WorkArea/DownloadAsset.aspx?id=10...
"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)