Page flipping was probably the biggest advantage Apple II had over CGA, and CGA wasn't the greatest possible, but the comparison isn't quite so simple as Apple II had more colors (which isn't always true).
First, do you have the CGA card connected to a composite or RGB monitor?
If a composite monitor, CGA wins big time in number of colors. You could get 16 at a time (even more with some programming trickery, though I've never seen anyone really take advantage of this). Also, there were quite a few different palettes available, you didn't have to always have the same set of 16 colors. Resolution is slightly lower (160x200 vs 280x190) though. A couple years later Apple introduced the double hi-res mode which would bring 16 colors to the Apple II.
If an RGB monitor, you can get 16 colors in 40 or 80 col text modes (compare to black and white on Apple, plus 80 col text was much sharper on CGA), but only get 4 colors at a time in 320x200 graphics or 2 colors in 640x200 graphics. Resolution is higher (320x200 vs 280x190) than Apple II. Also you can make each pixel whatever color you want without affecting other pixels, something not possible on the Apple IIe due to the oddities of using a composite display and the manner in which Apple II created some of the colors. While some clever programmers made this an advantage to apparently gain two more colors (yellow and pink if I recall) there were certain pixel combinations that just weren't possible without some bizarre unintended consequences. You were not limited to just cyan/magenta/white/black as you implied, this was the default, but other combinations were possible; background color could be any of the 16 colors, and 6 foreground color palettes were available.