Why does a camera flash need to happen at the moment of the picture?

The timing of the flash should be while the shutter is fully open, because you want the subject to be fully lit.

It does no good to flash after the shutter closes or before. It would “black” or dark.

Yes, I get that, but is the method an echo of the old pyrotechnic flash? Why not have the light going before during and after the shutter release to make sure the light is right? Digital cameras tend to have an unpredictable effect compared to what’s visible on screen pre-flash.

Yes, it’s because the developers think people expect a “flash” feature, and they think they can have it send light only when it makes sense.

You are correct that it would be a nice alternative feature, to be able to illuminate while the human on choosing to take the picture.

Sadly, even if I turn on the camera’s light, as soon as I activate (or task switch to) the camera, my “mobile device / smart(ass) phone” automatically turns off the light. I doubt there is a real good reason for that.

I have seen some people say there are some camera-phones that don’t turn off the flashlight. Also it seems entirely possible to write a program that will do so. There should be a setting for it, somewhere . . . but that doesn’t mean there is.

There are couple of reasons.
A camera flash is usually very bright for a short instant. It can be as short as 1/50,000 of a second to about 1/100 of a second. It takes a lot of power to make something that bright. That power is usually delivered by discharging a capacitor that has been charged by a battery. A capacitor can supply many amps for only an instant. If you wanted a flash to be that bright for a longer time the electrical components would become huge. Bigger capacitors, bigger batteries and a greater need to charge.

A matter of exposure control.

Modern cameras with exposure controls (made after 1970) have a sensor the reads how much light comes back in to the camera. I had it on my 1974 Nikon Photonics 35 mm.

The meter stops the flash after reading the the scene has been fully lit. If one takes a flash picture of a landscape, the flash stays on longer than if you are in a studio with auxiliary lighting.

The flash is triggered by the action of the shutter. The flash may actually be at it’s brightest when the shutter is fully open, and that is all that matters.

Agree with @zenvelo – despite the short duration of a flash, the camera can command it to stop sooner than the default duration.

Note that “red eye” reduction is the use of flash for a fraction of a second before the camera shutter opens. This causes people’s irises to narrow down so that less light from the flash gets into their eyeballs and reflects out. Red would have been the color of people’s retinas, as seen by the camera.

There is a type of photography used to detect the light emitted by an object or chemical after it is stimulated by a certain wavelength of light.
Here is the timing order:
The shutter is closed.
The flash fires.
The flash stops.
The shutter opens.
The shutter closes.
This cycle is repeated at 10–30 flashes per second.
It is often coupled with a headset that uses a blanking screen like a welder’s automatic visor that is synched to the flash. The screen is black, opaque, when the flash fires and turns clear when the flash is off. That way you can view what the camera will see.
This technique is sometimes used in forensic cases to find traces of blood

If we’re talking about phones LED “flash” I’d guess the main concern is power management and not annoying your subject by shining a very bright torch in their eyes for longer than you have to. Otherwise all your photos will be of people squinting, turning their head and blocking the view with their hands.