The usual solution to control lots of RGB LEDs is individually addressable LEDs like WS2812B. I believe these have a PWM frequency of 400Hz, so perhaps addressable LEDs based on a chip like GS8208, which has a PWM frequency of 8kHz, would be preferable.
Since each LED has its own free running oscillator, they won't be synchronized. The frequency of each LED will drift randomly. They don't turn on and off at the same time, which means the PWM flicker of all the LEDs is smoothed.
8 kHz PWM produces absolutely no visible flicker, and these LEDs are available on readymade strips, so you should really try that first.
If you still have flicker problems, then the next easiest solution would be to use a readymade LED strip with control chips like WS2811 and solder capacitors in parallel with the LEDs to smooth the current. This won't work with GS8208 because of its particular output structure which puts all the LEDs in series to get higher efficiency when a white color is selected (and also allows higher supply voltage).
But with a chip like WS2811, or perhaps one that has higher PWM frequency, each LED is connected independently to a PWM current source. This means efficiency is worse than GS8208 for white and better for single color, and supply voltage has to be 5V, but... adding capacitors could work.
The next solution would be to control each LED individually, which would indeed require tons of components. You'd have to use linear current regulators, because it isn't realistic to use switching current regulators due to component count. These would need to regulate the current, which means you'd have to add current sense resistors and switching current regulation chips.
You could get almost the same efficiency by using just three voltage output buck converters, generating just the minimum voltage for each color, and using that as power supply for each color. Then the linear current driver would use very little power.
But you still need one DAC channel per LED (3 per RGB LED) which may be filtered PWM (add one resistor and cap) and a cheap voltage controlled current source, ie an opamp, FET, and resistor. You could make it with just one transistor or two, but then it'll have substantial offset voltage, which means you won't know where the zero brightness point is.
So really, try it with GS8208, see if 8kHz PWM frequency does the trick first.