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.
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.