SEA 3000, the programme that is to build Mogami-class frigates for the RAN in a new facility in WA that is at best, under construction now, was not announced until Feb 2024, just over two years ago. To look back at Australian shipbuilding programmes running a decade ago that NZ might have participated in, the only ones running AFAIK were the ones which led to the Arafura-class OPV and the Hunter-class FFG. Early 2016 would also be before the selections made/contracts signed for the two classes which occurred in 2018.
The other, very large fly in the ointment, would be that NZGov't appears to have been aiming for a mid-2030's replacement for their frigates for some time. Had NZGov't really wanted to replace the RNZN frigates on/around when they reach 30 years of service, then the first vessel should have been planned to enter service by the middle of next year. Instead, it looks like the planned for replacement is still nearly a decade off, with some of the planning now being done. Also, in order for NZ to have gotten a lead replacement ship into service by ~2027, NZ would likely have needed to start working towards this in the early 2010's during the Key gov't.
Whilst looking back, this might seem now to have been appropriate back then, I do not think at the time members of gov't would have been willing to entertain any notions of the RNZN purchasing new frigates.