As a kiwi who has spent half my life living in different countries, I can say trust in the government here, while not universal, is higher than anywhere else I've been. I think most people would do it
I can already see the cracks appearing as some idiots try to score points or promote their own agendas and the media are happy to give them air time, the "covidcard" or phone tracing app will open another one of those cracks even though like you say the majority would conform
The hybrid approach the article suggest where perhaps the card + app could be compatible would be ideal.