But that means it has reduced refresh rate.

The two are not related at all. Refresh rate is how fast it can accept input, whereas this is how fast it can do TDM of colors and intensities

How quickly? Surely well above 1 kHz (1000 FPS). Otherwise you will see flickering.

According to this, humans can't see flicker above 100 Hz for most smooth images, but if the image has high frequency spatial edges then they can see flicker up to 500-1000 Hz. It has to do with saccades.

https://www.nature.com/articles/srep07861

Single chip DLP projectors strobe red, green, blue, white sequentially. Modern DLPs use separate light sources (LED/Laser) and pulse them at a high frequency - kilohertz I assume. Before we had high-power LEDs DLP projectors used a xenon lamp and a color wheel (https://www.projectorjunkies.com/color-wheel-dlp/) spinning at as little as 60 revolutions per second. This caused a "rainbow effect" which was very annoying to some people, but apparently enough people didn't notice it that those products got sold anyway. So somewhere around 180Hz is the bare minimum.

You don't need all the colors. As every household white LED bulb proves, you can get it with just a combination of blue and yellow.

You'll get atrocious CRI/sick skin tones that way. There's much more fleshed out spectrum in nowadays LEDs, especially warm white variants.