I can envisage the latter - that there are good reasons for a resistant spore stage in many organisms' life cycle, and radiation resistance is important for anything that might get cycled through shallow near-surface ice.

The former is more tricky. Sure, you get to colonise a whole new moon, but once that's been done once for each moon, the game is over. And I'm not sure there's much "benefit" for the parent that does it the first time, either. But I need to think on this - it's an interesting idea.

The "intermediate forms" business is tricky. You have to try and pull back from an almost irresistible human tendency to assign volition to evolutionary processes. A spore that evolved for the reasons you describe (radiation resistance, durability, spreading across a moon) might turn out by complete luck to be a good way to spread across space. After this had happened by accident enough times, spores that were tougher would tend to spread further. After even more time had passed, the adaptations might begin to favor those particularly suited for space travel.

The "final" species would not necessarily supplant their ancestors. Back on the original moon, maybe the first ancestral spore species would still be doing quite well on that moon only, being well-suited for survival there. But on unpopulated moons, the only species you would find would be the best space-traveling species.

Frankly, I don't think there are enough potential environments around a single gas giant to make this work all that well. Now, if you start postulating little solar sails on seeds that originally evolved them to ride wind currents... ;)

For all we know, coconuts first evolved because they needed a tough shell to protect the seed from some kind of animal. The rest was serendipity. It always is. We see the current coconut palms all over the world because that's the species that came up with the best way to spread itself. I'm no plant expert, so I don't know if some small isolated coast somewhere has the ancestral palm, with a nut not quite as well suited for ocean travel, still doing well in its home environment.

Lucky for us it's such a small step from a hand designed for grabbing a branch to an opposable thumb.