The terrible part to me is them saying "the majority of cases are resolved within 24 hours", this should be like a 15-minute delay at most when you can present your passport with its number which also matches your biometrics (as would be the case with twins).

I myself recently had a 10-minute or so delay due to a visa issue. I was departing from Johannesburg and was going to use my Italian passport for entry into New Zealand. I had acquired an NZeTA visa about 15 months before but didn't think to consider until checking in that it was against my previous passport number which I had since renewed.

When the agent said they were just having to double check my visa (while my wife's and children' were fine), the fact the passport had been renewed occurred to me and I told the agent about it which probably saved a couple of minutes of them having to work out this was the issue. They phoned New Zealand, explained the situation and within about 10 minutes the New Zealand side updated their system and I was able to complete my check-in.