A French threat prompted the signing of the Treaty

The notion that a possible French threat to take over New Zealand encouraged the British to seize sovereignty of the country does not match with what is known of British motives for the Treaty. France's flirtations with New Zealand in the early 19th century never reached the point of penetration, and French officials were ambivalent about the opportunities New Zealand offered them. At the same time, the British knew that their Gallic neighbours had little inclination and less chance of making New Zealand a French possession. Indeed, a cynic might suggest that the French were a far greater threat in New Zealand waters in the 1980s than the 1830s.

I don't agree with that. The French sussed out New Zealand and travelled back to Akaroa, in the Sth Island to fly their flag, and claim sovereignty. The British arrived 6 days before them and beat them to it. There is a monument in Akaroa which attests to this fact.

My ancestors were some of the First French Settlers to arrive in Akaroa, NZ in 1840.

The street signs there are in French... it was more than a flirtation by the French with NZ, as mentioned in the quote.