The facts, as I see them, are pretty clear.
The OP’s family did NOT have valid ESTAs when they got to the airport and tried to check in. The CBP says in that situation you will likely be denied boarding, which is exactly what happened.
The OP did subsequently obtain ESTAs, before check-in closed, but VS apparently didn’t receive the marker from the CBP required for travel. The OP claims VS refused to call the CBP to seek permission to fly until it was too late.
So: could VS have done more to help? Maybe, but is it obliged to? That’s the question that will decide whether the OP can claim compensation for involuntary denied boarding.
The OP’s family did NOT have valid ESTAs when they got to the airport and tried to check in. The CBP says in that situation you will likely be denied boarding, which is exactly what happened.
The OP did subsequently obtain ESTAs, before check-in closed, but VS apparently didn’t receive the marker from the CBP required for travel. The OP claims VS refused to call the CBP to seek permission to fly until it was too late.
So: could VS have done more to help? Maybe, but is it obliged to? That’s the question that will decide whether the OP can claim compensation for involuntary denied boarding.