I live in NZ as well - forests provide a myriad of benefits, the ones you'd probably consider most importation is the sequestering of co2 and the production of oxygen (that thing you need to live). Beyond that they are a habitat in which much wildlife finds a more suitable environment (forested areas are protected from birds of prey), act as wind breaks across areas such as the Canterbury plains, help with retaining topsoil and so much more.
The shade they provide your BBQ is literally the least important role forests play.
The o2 production that forests and rainforests provide always seems a bit overstated - your surrounding oceans are even better at that, but algae don't help with regional smog much, do they?
Carbon sequestering, wind breaks, increased biodiversity, topsoil protection, game reserves - they may not seem huge to the average person, but like you say they are crucial for our continuity.
The shade they provide your BBQ is literally the least important role forests play.