I must say that I have never quite understood negativity towards United on the Aust-US route, which they have flown for over 20 years.

I have flown United to the USA and back every time (from 1990 to today), a total of 7 times with number 8 already booked. Whilst they are not perfect, I am not sure how good a 14+ hour flight in economy can be, whatever airline you fly. Whilst the lack of a personal entertainment system seems to be the main negative point raised, I have never felt deprived by not having it in place of the main screen (I was not that impressed when I had one anyway). Further, on such a long flight leaving in the afternoon/evening I often end up sleeping for a significant portion of the flight, even when I want to see a movie.
Also there are a few lesser known benefits: -
Often overlooked is that for one annual fee, currently US$425, you can upgrade all United tickets for you and another person on the booking to Economy Plus giving you both extra leg room.
You can see the seats available before you book, and choose and change seats for FREE online any time after payment.
The Mileage Plus points you accumulate are a nice bonus.
Overall the money you typically save by flying United, particularly for a family, is worth it and much better spent during the holiday itself than on the flight getting there.
As for your specific situation
becpee, I would recommend booking the entire row of three seats on the right of the aircraft so you are all together and have the window and aisle to yourself. Have a great trip, whatever airline you fly.