to make sure that I'd have the visa in place for my first day
Sorry, but you CAN'T be sure of any such thing. Getting a visa is not like buying a commodity. It's a request to the NZ government to be admitted on certain grounds, and every piece of evidence in your application has to be assessed. Also, your application joins the queue of everybody else's applications, and they're dealt with in the order they arrived, regardless of what an employer and applicant may wish to have happen. Any statement of times by INZ is NEVER a promise, only their best guess based on recent averages.
WTR and Essential Skills visas are both temporary work visas. If one kind wouldn't be ready in time, there's no reason to suppose the other would be. In fact, there will be less checking to do on WTR than on Essential Skills.