Do RGB LEDs actually behave like three separate LEDs?
Yes, but they are not full separated LED's, as @Joby Taffey stated they share a common anode or common cathode. In order to use the circuit you said, you'll probably need to get one common anode one and connect the CA pin with the PWM output. The other 3 pins (cathode for R, G and B) will be switched with an transistor for example.

You could use an common cathode, but in this case, with an PWM duty cycle of 100% (turned one) the corresponding led would be turned off.

One problem I'd see is the refresh rate. You can't turn on the 3 leds at once because you want to control each color individually. The solution is multiplexing. You should use an enough high PWM frequency and enough low multiplexing frequency, so the multiplexing will not interfere with the color intensity while trying to avoid flicker with multiplexing.

And keep in mind that the output luminance will be only one third of the tri-led maximum.