Okay, I like having a culture in development. Maybe some rough equivalent of the Renaissance? See more below.

Your last idea could be interesting b/c we could have things that initially support either viewpoint: A religious devotee encounters one of the entities causing the illness or some phenomenon related to their presence, while a scientist makes some connection to radiation poisoning and the makeup of the nearby star. The revelation later is that these beings usually exist in the sun's atmosphere and so are saturated with the radiation, perhaps even need it to thrive, but some rare solar event (A flare or eclipse? Perhaps this is also something the religious people interpret and celebrate) inadvertently redirected them onto the planet. Oh, plus we can have the threat of a good ol' witch-burnin' that the [i]Enterprise[/i] has to try and prevent.

Although, a lot of this behavior sounds like what you'd expect from a pre-warp drive people, which would put the Prime Directive in effect. So maybe this culture is descended from an expedition sent out by a warp-capable race that got marooned on the planet and lost much of their knowledge? Picard & co. might waffle a bit over whether this gives them leave to interfere or not (I don't know if there's some precedent on the show), before circumstances possibly force their hand (see following).

Additionally, to borrow from your second idea, maybe the survivors of the crash were deliberately manipulated by someone who's become a venerated, saintlike figure in their present-day religion, or even the god they worship? And so the higher-ups of the religious order know full well they came from another world and some other details about their origins and former capabilities, but the general public doesn't. These could be the same folk who 'force the crew's hand' by way of hailing them with secret comlink technology. Members of the crew might then have to go planetside to investigate, complete with surgical alterations or disguises and are viewed as foreign scientists or hermetic healers.

Hopefully that doesn't all sound too complicated...

EDIT: Aaaaand a good chunk of that seems reminiscent of "Thine Own Self," not to mention "Who Watches the Watchers," but it still has potential.