Happily, I think I’ve figured it out. Yes, a comet is a “snowball in space.” But a comet isn’t just formed when liquid water freezes, as it often does on Earth. A comet is formed when a body gets so cold, the air on it freezes. The nitrogen, oxygen, hydrogen, etc. This only happens very far from the sun–so comets can only form far to the sun. If they formed close-in, their gases wouldn’t freeze.

So long as a frozen body stays way out there away from the sun, we’re not going to see it. It’s only when comets come closer to the sun (say, by getting knocked out of their original orbits,) that their gases begin to sublimate under the sun’s glare and they appear as bright, fiery comets in the night sky. Then, if it is lucky, the comet swings back to its frigid neighborhood before it totally melts away.

This explains why the comets we see have such eccentric orbits–the eccentricity allows the comet to freeze, sublimate, and freeze again. Without that orbit, no bright comet.