Circuit to drive 3 LEDs that flicker independently
The basic idea behind the article is sound enough- don’t try to develop the flicker algorithm but simply amplify the current from the off-the-shelf (sort of) LED. Without some characterization of current vs light output of the LED it’s difficult to judge what circuit would be appropriate. You could measure that with a digital oscilloscope and a bit of circuitry. It would be easy to simply build the suggested circuit and see if it has any defects in the appearance. 

An alternative would be to develop (or copy) a flicker algorithm and implement it in a microcontroller, using PWM and a MOSFET to drive the LEDs. Many years ago, I did that with an 8-pin PIC microcontroller and it worked out okay, but not especially convincing. I also mixed colours to get blue and orange flame-like hues.