Hi Julie,
No, they're not complete, but missing records occur more often in the early days of the Colony, rather than later on as bureaucracy developed.
I don't think your individual would need to show a birth certificate for either a marriage or on joining the army (the latter only wanted men who could fight, and there are many examples of men who 'lifted' their age for WWI and if they survived to join up for WWII 'dropped' it again, neither of which might be accurate.
More likely is that he appears in the NZ BDM indexes under a mis-spelt name.
Regards, riley