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.
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)
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.
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...
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.