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:
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.
I believe they had the standard CMYK, four shades of black, as well as red, orange, green, and either violet or blue. But it has been a bunch of years so this is off memory. I honestly don’t remember the name of it. What I do remember is that they didn’t not have a web based ordering system. Instead they had a piece of desktop software you had to install. And you had to prove that you are a professional photographer before they would let you create an account. I am not a professional photographer but I did enough amateur photography that I managed to fake my way into it and placed a few orders. Quality was definitely better for all options compared to Nations Photo Lab but so was the price and the ordering setup was much more complex so I didn’t continue using them. They did have a lot more specialty options than any other printer I have seen.
"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)
I paid for college in part by doing digital prepress. We had CMYK and 8 and 12 color separations.
CMYK always has a dramatic color shift from any on-screen colorspace. Vivid green is really hard to get. Neons are (kinda obviously) impossible. And, hilariously/ironically (given how prevalent they are), all manor of skin tones are tough too.
Photoshop and Illustrator let you work in CMYK, and is directionally correct. Ask your printer if they accept those natively.
My fav part - if you're preparing an ad for a newspaper you need to contain the sum of all of your CMYK components to under 120 or so value otherwise the print will either dissolve the paper and it will go through.
