I'm sorry that your ex-partner is in a difficult situation, but it happens to a lot of people. The law is hard, and doesn't take account of decent people being uncomfortable because of it.
This doesn't make sense to me!! How can they possibly expect someone to just stop working for their employer for up to 30 days (or however long it takes to process a visa) when they have to have money for food and bills etc!
It doesn't have to make sense. It's the tail end of the law that doesn't allow anyone without a visa, e.g. a years long overstayer, to work. Someone on an interim visa can go on working if they're between two visas of the same sort - that is, if they'd already got a work visa for a job on the grounds of the work. But your ex-partner had his previous visa on the grounds of your relationship, that is, as a favour to you, a NZer, and his necessity to the NZ work-force (or not) still has to be investigated.
The other thing that I am confused about is the fact that he already has a job and now by applying for the Essential Skills Work Visa - they want him to get a job offer? But he's already got the job? How is that going to work? Does his boss have to advertise the position now?
As just said, he had his open visa because of you. Now the employer has to show that there isn't some native NZer or person with a Residence visa that could do the job.