Delayed in LAX

After a 2.5 hour delay for my flight from SF to LA, I briskly walked 1.5 miles to get to the international terminal in LA to go though security. Only one scanning machine was in use so it took me 45min to get to the front of the line which left me 30min before my NZ flight departed at 9:30pm. Unlucky for me, my bag was flagged by a securty official and I had the joy of fully unpacking my backpack with security and a tight timeline to make my flight. As expected there was nothing out of the ordinary in my bag so I repacked and ran to my gate which had already begun the boarding process. The check-in agent informed me that I needed a printed copy of my flight leaving NZ before I could board the flight entering the country. I had booked a flight to Sydney earlier in the day on my iPhone and only had an email receipt of my flight, no hard copies. Whitney from NZ air quickly took me to the sky lounge to print my Sydney ticket warning me that I would likely miss my flight but could be booked on the next flight one hour from then. My brother George was scheduled to fly out on the next flight so it was fine with me if I missed my flight. After printing 2 copies of my ticket to SYD, it was 55min until the next flight and George had still not landed on his connecting delta flight and had not checked in. The check in cut off time mandated by security is 45 min before the flight departs. George's flight to LAX finally landed(one hour late) and he ran to the international terminal to check in. Unfortunately, it was 8min past the check in cutoff time so he was not allowed on the flight. The next available flight was not until 1pm the following day and it was only flying to Auckland, not Christchurch. We will have to spend a night in Auckland before flying to Christchurch to Start our travels. 2 days of traveling down the drain, thanks Delta.