Are there any downsides when painting with a cyan-magenta-yellow palette instead of the traditional yellow-blue and-red?
Cyan, magenta and yellow are actually the primary colors you should use when painting with inks and paints.
(I'm surprised we didn't have this kind of question asked before, because most of us are taught the wrong primary colors in school and by paint manufacturers)
Fair warning: the information in the following section is WRONG but it's what most people learn about the color wheel.
The typical color wheel that is printed in (some) art books and on paint boxes looks sike this:

The primary colors are: red, blue and yellow.
Every child learns that:

Mixing red and green yields brown
Mixing red and blue yields purple
Mixing blue and yellow yields green

But have a closer look. The wheel is not uniform and yellow has a much bigger area than all other colors. The color right next to red is magenta, which is actually lighter than the two colors it's supposed to be mixed of. And anyone who ever tried mixing a vibrant purple from red and blue was disappointed by the muddy outcome.
How it actually works
There are actually 2 sets of primary colors and both of them are correct, but they apply to different ways of mixing color.
White light is a composition of every visible color. If you shine a red light at a spot, then add a blue light and finally a green light, the center where all colors add up, is white. This is called additive color. This applies to all TV, phone or computer displays and anywhere where color is made by mixing light.
If you mix red, blue and green paint on a paper, you end up with (at least almost) black, though. Your paints don't work the same way as light. You need to take color away (subtract from the mix) to end up with white. This is called subtractive color. This applies to all printers, pens, inks and paints and anywhere where color is made by mixing substances.
The primary colors of additive mixing are: red, green, blue. (RGB)
The primary colors of subtractive mixing are: cyan, magenta, yellow. (CYM or CYMK)
The primary school art class colors red, blue and yellow aren't even a valid set of primary colors!

(Image source and more detailed information)
And here is another example of CYM color wheel that displays how adjacting colors mix:

Can only 3 colors suffice?
Unless you use high quality inks, the answer is still: no.
The color wheel is an idealized model based on wave lengths of visible light. We can manufacture color filters that absorb all but a single wave length (color) and mix colors additively in a perfect way. But as far as I know there are no pigments that reflect only a single wave length, which means that there is no paint of a perfect primary color.
The chemical and physical composition of pigments and the binders and additional ingredients in paints influence how the paints look when mixed. Linseed oil, for example, has a slightly yellow tint, which always influences the oil paints it's mixed with.
Many paints, especially cheaper ones, already contain a mix of different pigments. They may look like a primary color to your eyes, but they physically react different than a pure primary color pigment when being mixed.
Most yellows, for example, contain chalk or white pigment to make it more opaque without adding more of the expensive yellow pigment. You'll never get a rich, lush green by mixing a cheap yellow paint with blue.
(Side note: that's why the ink for inkjet printers is so obscenely expensive. I dare say that inkjet printing is the closest we've ever come to perfect subtractive color mixing, and it still isn't perfect at all. A printer has to use very pure, high quality ink to be able to mix all colors. And even then you need an additional black ink cartridge because the mix of all 3 primary colors doesn't yield a perfect, opaque black.)
In addition to that, it's simply more convenient to have certain colors premixed, because it can be very tricky hitting just the right hue when mixing a color from just the primary colors.
And lastly, it can be a matter of cost. Especially brown mineral pigments like Umber and certain chemical pigments are cheap to manufacture in high quality, so mixing the same hue from more expensive primary colors is a waste of resources.