"Why, when they were so strong, did Maori invite the British in?"

Sources here.

"The view that the Treaty has become the straitjacket of Maori history is the starting point for the present study. ...
    "When Europeans stepped into the southern dawn, the people who had thought of themselves as constituting the whole human world found that they were one of its fringes. ... Assertive and risk-taking in cultural personality, Māori fought to eradicate their fringe status by pursuing modernisation, including the political modernisation that would create a path to the Treaty ground. .... The question then becomes why, when they were so strong, did Māori invite the British in? ...
    "Because Māori society [had been] organised around the chance of war, the initial effect of the introduction of muskets was indeed the expansion of tradition. Serial wars followed from the idea of their possibilities. Soon enough, however, war became an agent for its own collapse. Nga Puhi’s raids were predatory larks, fought for neither territory nor strategic advantage. ...
    "[T]he terror of the gun [had] caused social disruption analogous in principle to that of current world ethnic strife. From Auckland to Whangarei was empty. The Hauraki tribes fled inland, and Ngati Kahungunu to the tiny edge of their vast territory. Modern Taranaki was deserted, and some of its displaced people virtually exterminated the Chatham Island Moriori. Ngati Toa, forced into pre-emptive migration, ravaged the south — the list is representative but not exhaustive. Similar things had occurred in the history of most tribes, but there is no previous evidence of near-total war. A detached modern historiography lists battles, but makes the musket wars events without real effects. Yet the wars were a modern catastrophe for Māori, not a traditional one. ...
    "In the 1830s northern Māori sought meaning in their post-contact experience through understanding how the foreigners ordered their world. This was a period of rational and intellectual response to European culture in which Christian teaching became a political primer for change. Consciously replaying the conversion of the barbarians, the missionaries taught that peace was the condition of political and social modernity — that is, of a European-style society.
    "This impacted heavily on culture, because tribal histories were almost exclusively histories of war. Fighting was central to the social identity of Māori. [It set] up peace as the condition of modernity ...
    "Their attention to the missionaries, and subsequent support for a treaty with the British, was not without history, but a response to lived change. By this reading, then, a possible basis of Māori citizenship was rational choice.
    "The rationality of the chiefs has been obscured by the rationality of the British side of the Treaty, which entirely dominates the literature."

~ Lyndsay Head, from her article 'The Pursuit of Modernity in Māori Society'