Congrats on yoru job offer! I know nothing of the field but I imagine it's super competitive!
I've lived in NYC and I've also moved my kids at secondary age and that was within the bubble of international schools. It's hard!
Have you looked at high school entry in NYC? It's not like other places, you don't just rock up at your local high school. The best high schools are very vyer competitive and you probably have missed entry for your 14 year old. Of course you can live outside NYC in suburbs where schools are easier but then you have a commute (plus high rents to reflect the high property taxes through which schools are funded).
Also, have you actually calculated costs in NYC? Rent, health insurance, decent food, any exra curcciculars - they are all more than the UK. Health insurance in NYC is insane and you will have many health related expenses.
You then also need to consider uni entrance for your oldest. It's likely they will get home status if you can show temporary absence (which if you're on HIB visas and not got green cards is likely) but as a pp pointed out, not student finance which is based on actual physical residence.
I would recommend you take the role and have the family come over during holidays and get back to UK as much as you can. If your DH is hands on and a good Dad, this could work. They're old enough that you can message and videocall. See how it goes for a year and then decide.
I do think it was unfair of your DH to tell you to go for it and then change his mind but maybe while it was a hypothetical it felt different!