I'm going to answer this in the context of European settlers of NZ in the 19th century, rather than Polynesians in the 13th, because it gives me a chance to use my high school history teacher's summary of them as "weird". (Yes, it was a NZ high school.)
The vast majority of European emmigration in the 19th century went to the Americas. If you find yourself wondering "why didn't they go to..." then the average European emmigrant would agree with you. The people who choose a 100 day trip by sailing ship to the other end of the world were the weird ones.
The first European "settlers" in NZ were whalers and sealers who were dropped off by ships to spend some months whaling and sealing, then to be picked up again, or re-supplied, all going well. These were soon added to by camps cutting timber. Virtually all of these Europeans were men and a number married local Maori women, and kids, and became more settled. Maori were eager to trade with Europeans, in large part to get their hands on muskets. A few traders drifted in over the early 1800s, and some Australian convicts, either runaways or discharged convicts.
The first missionaries arrived in 1814, on Samuel Marsden's mission. This was followed by several more missions. The missions were on a significant scale, including carpenters, and a number of married European women.
The first planned European settlements of NZ were by the New Zealand Company. Its founder Edward Wakefield wanted to create a better England in the colonies, and NZ was the place to try out his ideas. These were funded by selling land parcels to investors (many of whom never visited NZ), and the money used to subsidise migrants' passages. Why did investors buy the land? There's little research on this, but presumably speculation: wouldn't you like to have bought a piece of Manhattan in 1700? Why did people actually emmigrate? The Company painted a very rosy view of life in the new NZ colony and of course most of the new settlers were disappointed, many leaving again. The weirdos stayed.
There were several other settlement schemes for NZ that bought some people out in pursuit of one idea or another.
The first big surge of European migration was the Gold Rush of the 1860s, first in Otago in 1861 then the West Coast. Many of the miners who rushed in rushed out again but a number stayed. People started to figure out how to make a profit from sheep farming in the lower South Island over the 1850s to 1860s. Local governments subsidised some immigration but the next big push came in the 1870s when the NZ premier Julius Vogel subsidised immigration on a national scale, particularly of farm labourers and women. Immigration dropped off during the wide spread depression of the 1880s but picked up again in the 1890s as people worked out how to make money from dairy farming and processing and coal mining and similar activities.
So why did Europeans come to NZ? They were the weird ones, the ones who thought that there was the chance of making a fortune, or the weird ones who believed the New Zealand Company's advertising. It wasn't entirely weirdness, of course, government subsidies for migration reduced the costs for many and the NZ climate lacks the extremes of heat and cold that North America has, and the NZ fauna isn't as murderous as the Australians. But, at the end of the day, we are talking about a small proportion who decided to make it this far. Most people took a much shorter trip.
And I'm sure our ancestors would have been even more motivated if they had known that in 2019 their descendents would be voted world's sexiest accent.
Sources
"Dicky" Barret biography, https://teara.govt.nz/en/biographies/1b10/barrett-richard
https://nzhistory.govt.nz/culture/home-away-from-home/sources#findings
A historian might challenge your view of Australia - convicts are really only important to Sydney and Tasmania, and most convicts didn't tend to think of Australia as a hellhole.
Sydney, most of Tasmania and Brisbane (the foundations of NSW, Tas and QLD) were founded as penal colonies, whereas Perth, Melbourne and Adelaide (the foundations of WA, Vic and SA) were settled free. Sydney and Tasmania ended their transportation around the 1850s, which was when Perth began accepting convicts, twenty years after settlement, for the cheap labour they provided. WA ended transportation in 1868, after about 20 years, and convicts are rarely ever mentioned here except when discussing the building of colonial architecture and infrastructure, like the Perth Town Hall or Fremantle Prison. Brisbane was founded as a 'worst of the worst' style settlement, as were places like Port Arthur in Tasmania, but they quickly transformed as Sydney's convicts stopped being shipped to them and free settlers demanded rights to settle. Melbourne used some of NSW's convicts very briefly, and Adelaide remained an entirely convict-free settlement, as did every other major urban centre, most of which developed well after the convict era, like Darwin and Canberra.
With each colony having developed independently, with unique challenges and solutions, cultures and politics, enormous geographic divides between them, and constant waves of free settlers, like those of the gold rushes, it is hard to argue for Australia having been founded in/from Sydney, by convicts. Convict history is really only important to Sydney/NSW and Tasmania.
A good resource: the Origins of Australia's Capital Cities, edited by Pamela Statham
One could also challenge the idea that the convicts were unwilling/unhappy participants. Most enjoyed life in Australia, and many became extremely wealthy. Nobody wanted to get whipped and shipped, and those on the First Fleet with Arthur Phillip certainly suffered as Sydney was first being established, but life in Australia could be infinitely better for a poor/working class Briton than back in the UK. For example, as prisons became more severe in the UK, that was generally resisted here - in fact, reformers in Britain claimed Sydney didn't work as a prison, because it was essentially a reward, and thus didn't discourage repeat offending. Most of the worst/harshest prison management styles came to Australia as commands from British authorities, and were usually abandoned fairly quickly. Skilled and industrious convicts were often in high demand, becoming wealthy, famous and influential, and class was less of an issue in Australia, as were nationality or religion. Pardons were granted freely and convicts even successfully striked to have their afternoons off most days.I'd argue that convict life has been heavily dramatised to sell books and tours, or make films and tv shows. Life could be harsh (nobody wants to get flogged), but a lot of the lifestyle is edited out to fit a preconceived notion of convict life.
A good resource: John Hirst's Australian History in Seven Questions.
Part of the answer can be found in what /u/LordHussyPants says here. /u/CrossyNZ also hosted a thread here that may be of interest about "Waitangi Day" (6 Feb), which is when the British government signed a treaty with most of the Maori (but not all). The question has been asked before, with an answer by /u/mormengil.