I had a flight cancelled in the great radar fire debacle of 2014, and the new flight they booked me on didn't have flights together for me and my two girls (2 and 8 at the time). I asked United to move us to be able to sit together, and they said they couldn't. The best they could do was tell me which of the three seats had a single traveler next to it, so I could ask that person to switch. Luckily, the kind person agreed to switch, so my 8 year old sat alone, and I was able to sit next to the two year old. But honestly, if I couldn't have been seated next to her, I would still have gotten on the plane. I'd rather not be stuck in the airport for two days, and if somebody didn't want to switch with me, then they'd get to sit by my toddler. I've actually had to ask people to switch a few times because of flight cancellations. It stinks, but I had no option.
That's why it drives me batty when people say, "Oh, I'd never switch with somebody." "People should check their reservations and seat assignments." "Don't be too cheap to pay for seats together." and so forth. Sometimes you can do everything right, and the airline still screws you over and you get stuck asking people to move. Just try to have compassion for other people. Luckily, I've found that most passengers are kind and caring and willing to help.
What I don't understand is why this family didn't board the plane and ask someone to switch. I'm pretty confident they could have worked it out to have adults next to kids. That would probably have been a better option than throwing a tantrum in the gate area.