Looking at this map and this map, it seems that the Chinese varieties spoken closest to Taiwan are Hakka, Southern Min, Puxian Min, Eastern Min, and maybe Wu. I assume this spatial distribution held the same when Taiwan was only first settled by Han peoples.
It would make sense to me that languages closest to Taiwan would have diffused most intensely into Taiwan, but this is not the case--Puxian Min and Eastern Min are not spoken except in Wuqiu Township of Kinmen County and Lienchiang County, which are extremely close to Mainland, and no variety of Wu is spoken at all.
In particular, it surprises me that Puxian Min and Eastern Min have not caught on in Taiwan but Hakka has, even though the Hakka homeland seems to be further away from Taiwan than Puxian Min's and Eastern Min's are.
Why?
Thanks.
For the reasons you mentioned we can disregard Wuqiu and Jinmen. They're part of Taiwan but as part of the ROC's Fujian Province, not the ROC's Taiwan Province. If we're talking Taiwan the political entity, then yeah Puxian and Mindong are spoken there, but if we're talking Taiwan the island, which is more what the question is about, then we'll ignore those.
Then what you have is basically a series of events going way back regarding Quanzhou, a Minnan speaking settlement, that gradually brought it to prominence
The answer to your question is in two parts. First, the prominence of Quanzhou (Minnan), and second, an otherwise urban/rural divide.
First, Quanzhou has been important for trade in SE China going back to the Song. It was the primary send-off point for trade with Southeast Asia. The population of the city grew significantly, as did its economic power. With more people migrating there and more opportunities for economic gain, Minnan speakers began to spread into other areas in SE Asia, Taiwan included. This is the reason for Southern Min being spoken so widely in the greater region.
Second, Puxian and Eastern Min are both primarily urban varieties, spoken in Putian and Fuzhou respectively, whereas Southern Min has Quanzhou but is also quite "rural". While Quanzhou had half a million people in the Song, it would be many hundreds of years before Fuzhou reached that. These were much smaller populations. When I say "rural" about Minnan, I don't mean that everyone outside of Quanzhou was a farmer, but the language group covered a much wider area. Thus as Quanzhou had become an important economic centre it provided opportunities for other Minnan speakers to take advantage of that and migrate, both to Quanzhou and abroad. However Putian and Fuzhou, with much smaller populations, much later economic development, and far fewer rural areas sharing a linguistic background, the effectiveness of large scale migrations was basically zero.
That's not to say migration didn't happen. It just happened under very different circumstances, and at a point where Southern Min was already well established on the island of Taiwan.
Put simply (tl;dr:), Minnan just had a way earlier and way larger head start, and thus a much more substantial population base to distribute.
Note: I'm about to travel w/o internet access for a few days so if you have follow ups I'll definitely try to get to them, but it might take a while.
