I agree. It works very well when the artist does it purposefully and can give really stunning results- [link] (This artist *TamberElla is particularly adept at using color)
Actually mixing warm and cool colors (such as red and green, or purple and yellow) is how you created "muted" colors, which is another way to tone down colors without using black/grey. If you ever work with a limited palette with real paints black is one of the first colors you take off your palette, since true black is very rare in nature. Most colors are a mix of warm and cool, like a perfect sky blue is blue with a touch of red to tone it down. This is more advanced color theory, however - your tutorial is a very good starter for folks, though, and probably more applicable to your style of art!
It usually works prefectly fine If you pick the right tones )