Why is flash sync speed much slower than flash duration?

According to What is flash duration?, flash duration at full power is 1/1000 s for decent flashguns. Presumably, at low power, the duration is even shorter. I understand that film era flashguns didn't do the dual flash (once for measurement, another for exposure) but simply turned the flash off when there was enough exposure.

However, my camera (EOS RP) has 1/180 s flash sync speed.

Why is the flash sync speed much slower than flash duration? I mean, if we can control the shutter precisely enough for 1/4000 - 1/8000 s shutter speed (meaning the accuracy should probably be below 10 microseconds or else we have inconsistent exposure), why can't we control flash precisely enough for 1/1000 s shutter speed?

Clearly, the issue cannot be limited speed of light, because speed of light is 300 meters per microsecond. Microsecond is accurate enough, and nobody is using a flash 300 meters away. (Ok, the signal needs to travel twice, once to the flash via radio/infrared/cable, another time from the flash to the object via flashgun output light, but even then the limit would be 150 meters.)

Is the issue related to the slowness of igniting the arc in the flashgun? So that one can control the duration of the arc very precisely, but not when the arc actually starts?

Answer

Cameras that have a mechanical focal-plane shutter have two curtains, a front and a rear. For longer exposures, the front curtain opens and starts the exposure, then the rear curtain closes to end the exposure.

The mechanical shutter is relatively slow, so to create a quick exposure, the rear curtain must start closing before the front curtain fully opens. So, effectively, a slit travels across the sensor exposing each section for the desired amount of time.

The sync speed (1/180 for your camera) is the fastest speed where the sensor is fully exposed. That is, a slit is not used.

Some speedlights offer a high-speed sync option. In this mode, the flash creates a quick burst of tiny flashes so that the sensor is exposed equally as the slit travels across the sensor.