The temperatures high up in the sky don’t have a smooth gradient. You’ve got one patch of air that’s, I dunno, 52 degrees and you’ve got a patch of air right next door that’s 2 degrees. When moisture floating about up there moves from the 52 degrees patch to the 2 degrees patch, it freezes out and falls as a raindrop (or hailstone), and anything that hits the boundary falls out basically immediately.
If temperatures up there changed smoothly rather than *BAM* abruptly, then rain would fall gradually as you describe. In some rare places in the world this does actually happen – for example, this is why England is under permanent grey drizzle rather than getting it all over with in one rainstorm.
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