Why does a prism spread its "output" out over space, not time?

A "time dispersed" signal is what the radar (sonar) engineer calls "chirp" or linear frequency modulation. It can be created by externally varying the frequency of an oscillator or by a medium whose propagation delay is a linear function of frequency. The former technique is employed by bats the latter is what a prism does. To see the effects of time dispersion you need time markers when you start and stop, say a short pulse, and a reconstruction mechanism of the dispersed pulse that results in a another pulse where you observe the ensuing delay. The bat's ear does the reconstruction for the reflected sonar chirp. For a dispersing prism a complementary "prism" that would reconstruct the original pulse you would need another optical gadget that would delay the various frequencies in the opposite way to that of the first dispersing prism so that the combination of the two would result in only a simple overall composite delay of the pulse; unfortunately there is no such gadget can be made of a simple glass having chromatic dispersion.

There is a famous technique to use pulse stretching-pulse compression for high power optical (laser) output generation, see https://en.wikipedia.org/wiki/Chirped_pulse_amplification

Huh. I never thought about that.

You just aren't fast enough to perceive the transitory time between when the first colour starts to leave the prism and the last colour hits the target. It is a beam after all and is identical everywhere in the middle so you can't distinguish any midbeam point from any other so so once the beam is bridging both ends it all looks the same. But you can observe the ends being different as long as they exist.

What you're asking for is a way for it to make sense. So you probably don't just want the equations that describe what happens.

Prisms work because they change the direction of light, and they change the direction more for blue light than red light.

Why do they change the direction? Maybe because they change the speed.

Imagine you're driving a car and you hit a patch of loose sand just on one side. It will make the car veer, won't it? The tires on one side are pulling you ahead more than the tires on the other side.

If you were just coasting along slowly and you hit a patch of sticky tar on one side, wouldn't that make you veer too?

So if light is like cars or bullets, particles that have a finite width, then going into materials that change the speed would change its angle if it doesn't go straight in.

And if light is like a wave, ditto. Imagine a wave like an ocean wave, with a crest. The direction the wave travels is the direction the crest travels. Imagine the crest going at an angle into a material that slows it down. The part that goes into the new material first gets slowed down first. The part that goes fast in the old material longest, goes farther. So the direction the crest moves through the new material will be different (and slower). The bigger the speed difference, the bigger the turn.