Firstly, I'm sorry for your loss.
If I were attending a one time event, such as a funeral, wedding, cruise, etc., I would allow an day extra at the arrival point. For cases just like yours.
Admittedly from you, the seats on Air New Zealand to LAX were not sufficient to accommodate all the passengers from the flight you were on. I don't know how UA handles their priority order but my airline gave First Class customers priority, then Platinum, Gold and Silver Elite. After that, it was first come, first helped. If the seats ran out, there is not much that can be done. Aircraft have limited space.
As far as the smiling agent, I won't comment other than to say it probably was a had to see it type moment to understand it. And getting a taxi paid for back home is actually pretty rare. Most airlines aren't in the habit of giving ground transport in the passenger's domicile.
You mentioned having to sit outside in the 'blistering' hot and having no suscreen or hat as you had to rush to the service. Here I have no sympathy for you as you had just mentioned you had to grab some lunch so there really was no rush.
Bottom line......you expected to be delivered to your destination on time and that is what should have happened. However, from what you wrote, UA did what they could with what was available. Yes, aircraft shouldn't break. I understand that but when they do, it's how the airline handles it that will truly decide what kind of experience you have. I saw, by your words, no rude agents, no lack of communication and no complaint by you of anything else. I think the UA agent did the best he/she could to get you there as expeditiously as possible.
Going home you mentioned a delay and the amount of time spent with each customer. Hmmmm.....that's called customer service and trying to please each one with their individual concerns. Some people complain when they are hurried along and forced into decisions and some complain when the airline takes too much time with a passenger. I wrote it yesterday and I'll write it again. Airline can't win for one or the other.
Hope your travels in the future are much less stressful.
