That is a true statement, but it misses the big point. (As the shutter would see it), it would simply become continuous light, like any incandescent light bulb (always On for the full shutter duration would be indistinguishable from continuous light). Like continuous light, there would be no motion stopping ability at all.

And even a 500 watt light bulb at 1/200 second shutter is 500 x 1/200 = 2.5 watt seconds of power input, also with very low output efficiency. The regular camera speedlight might be 75 watt seconds, with several times higher output efficiency, and vastly greater light ouput. And is fast, and is vastly more convenient to use than the 500 watt light bulb. Hollywood movies must use continuous lights, but they also have large trucks carrying huge power generators.

But your description is exactly what High-Speed Sync is (HSS), a "flash mode" optionally offered by some cameras and some camera flashes to mimic continuous light (to avoid sync issues). So HSS is a choice if you want to buy it, and want to suffer with its large limitations (of speed and power). But it is merely high speed sync (meaning a fast shutter speed can be used since there is no sync issue), but HSS is the full opposite of high speed flash. No shutter speed can be as fast as a speedlight can be. And continuous light lasting for that longer shutter speed duration would require excessive power (so HSS Mode typically must run at no more than around 20% power level of speedlight mode). Conversely, speedlights simply discharge a large capacitor as a very fast pulse. Which is typically relatively slow at Full power level, but speedlights are called speedlights because they are greatly faster at lower power levels.

The regular normal camera flash mode is called a speedlight (all camera flashes are speedlight type, but only a few studio flashes are) become even much faster at lower power level settings, perhaps 1/30,000 second at maybe 1/64 power. Which is ideal to stop motion like milk drop splashes or hummingbird wings, where the flash can be close to the action. See my site at https://www.scantips.com/speed.html

Flash is often somewhat faster than 1/1000 second, which has great advantage to stop motion, and to offer high peak power for an instant. Photographing the running playing kids when using a speedlight as bounce flash at say 1/2 power will be 1/1000 second duration, and will stop action quite well (even if the default shutter speed is 1/60 second). That assumes indoors, where the ambient continuous light is too weak to show any motion blur.