This story doesn't hold water since Expedia makes it clear seat assignments aren't offered with Basic Economy, and there is no commitment anywhere from Expedia to guarantee a family will be seated together. (A family sitting a few rows apart isn't ideal, but it's also not the end of the world. Perhaps the OP's family should broaden their horizons a bit and have some more faith in humanity to be good seat-mates to their younger children who have to sit in a middle seat by themselves, and also trust the F/As to do a good job of keeping the flight safe and secure.)
I recently purchased more-expensive tickets when travelling with 2 young children so the 3 of us could sit together. I would be displeased if I were asked to sit somewhere else so that someone who bought Basic Economy tickets and didn't bother to secure seat assignments could be seated together.
On a recent IAD-PIT flight, I did hear a good solution: the flight attendant announced that there were open seats in Economy Plus and she was ready and willing to swipe passenger credit cards if they'd like to set up here. Perhaps United should have offered that option to this family, and then given the passenger forced into a middle seat a refund or other compensation. Alternatively, United could have re-accommodated them on a flight at a later date that has more open seats.