> I would have thought that "Она ему нравилась" would be the one that means "he started to find her likeable and hasn't stopped yet (unless otherwise stated after this sentence)."
Imperfect form has no accent on the beginning (nor ending). We talk about middle of some continuous process. If we talk about past: "Она ему нравилась" we state, that some time in the past he was in the state of liking her. We do not know anything about start point of this process, nor about ending point. This "uncertainty" is implied, unless explicitly stated otherwise.

> So, would this sentence:
> 'она мне понравилась сначала, но я уже думаю иначе'
> Be better rendered as
> 'она мне нравилась сначала, но я уже думаю иначе'
> ?
No. Both variants are equal, because you have eliminated ambiguity by second part ("unless explicitly stated otherwise").
But "Она мне нравилась." vs "Она мне понравилась." have different accents. Without details we think that you have no reason to state that you liked her (in the past) except that you do not like her right now. Otherwise you should say "Она мне давно нравится" - ("I like her very long time"). But perfect form has accent on the fact that some time ago you _started_ to like her. It is the main idea of perfect form in this case and there is no another accents.

> Also, why is the original sentence not 'хотелось бы посмотреть, кто ей нравится'?
This form is absolutely ok and has very close meaning. You will not make mistake with your variant of translation.