Seems there are three questions here:
1) Is the OP -- larger island, settled by polynesians, isolated thereafter until James Cook -- plausible? The answers here have ranged from calls of ASB to claims of plausibility; I'm in the latter camp, but then I did post the thread here...
2) Assuming a positive response to the previous question, what sort of climate, polynesian culture, and the like, would develop on this island? Some good answers here, the island is sounding somewhat similar to New Zealand...
3) And this is the question I'm really curious about now -- under the same assumption, what would happen after Cook found the island? With the island itself, but also how that would affect the rest of the world?
All this seem about right? As to the third question:
I'd think Britain would like to add it to the Empire, seeing as it could prove sufficiently self sustaining; in effect, it'd be a lot like New Zealand's conquest. OTOH, this would take a little time, and the US would arrive on the scene in the meantime; and with the profits of the Old China Trade, they may take exception. So you might see a rebellion, similar to the Maori revolt, with support from the USA.