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If the temperature would drop 5°C both endemic unmistakable palms (including the most popular indoor palm of the world, howea forsteriana) would also perish. So, just to take these two examples, both the coral and the howea would have had to have developed from scratch during the last 10,000 years.

False premise.

Glad aout your comment. I was hoping that there would be a palm and climate expert to contradict this.
In fact using the climate data for Lord Howe station, located at sea level, they could still manage a 5°C colder climate.
The problem is that howeas can't survive on their own at the small strip of Lord Howe stretching at sea level. They don't tolerate full sun until they are adult. Their habitat is the very special rainforest on the slopes of Mt. Lidgbird and Mt. Gower (in the picture, as most of the time, under the clouds), 500 m above sea level.
And if you it is still possible to debate if there 5°C cooler temperatures would mean extinction, you should not have any doubts that a change in the rain pattern would!

500 km is not a particularly long distance for a coconut (if that is the "seed" type for this palm) to drift. If this island is surrounded by warm currents it is a fair bet that the water came from tropical areas rather than the ice caps. Same for the reef animals. Seems simple enough to me.