About the last paragraph of your post - I have yet to hear of any employer who took any notice of an LIA's being engaged when looking to take on staff. Does this company have an employment agency as a separate part of their business? If so, in your place, I would read the sales pitch on their website very carefully to find out how they go about what they do, and what they are promising, and expect to see some verifiable customer satisfaction letters. Also, have you looked to see that the LIA is licensed?
https://iaa.ewr.govt.nz/PublicRegister/Search.aspx
Many NZ employers are mistrustful of applicants from overseas, because so many turn out to be timewasters. People apply for a job without having researched the stringent NZ immigration laws, and so without realizing the amount of time and money it will take to obtain a visa. (And, as you will see elsewhere on the forum, there are currently considerable ongoing delays with the SMC residence visa process, with political undertones which have yet to be clearly explained.) Or they apply without having discussed the matter with their family, who are then not in agreement to such a change in their lives. In either case, the employer, who thought he had made an appointment for someone to start in the reasonably near future, would end up with nobody to do the job, and having to start all over again.
For this reason, many NZ employers are reluctant to deal with anyone from overseas, unless and until they have come to NZ to be interviewed. That demonstrates a commitment of time and money, and goes some way to proving the candidate's serious intentions to making the move. These days, it is very rare for someone to be offered a skilled job, the necessary prerequisite for being granted a residence visa or a temporary work visa, while still overseas, unless they have a VERY rare skill-set.
Quite a few members of the forum have managed to get job offers by making a recce trip, having contacted prospective employers or agents beforehand, telling them when they will be available for interview.