I'm sorry for the family to a certain extent but not 100 percent so as its the passenger's responsibility to ensure that all entry requirements are met prior to departure. I always check the Visa requirements as part of the planning stage before purchasing anything. I would wait to buy my ticket if I thought the Visa process would take time.

As,for the ESTA I would have dome a simple 2 second Google search and type in Entry Requirements into USA for UK passengers and applied for it before booking anything and wait for the approval. To take it a step further I would sign up anyone in my family who would potentially travel to the USA if I were not a US citizen.

For my trip to Brazil my Father and I applied for the Visa 10 months before our cruise.

At checkin I always verify to see that the boarding passes say DOCS OK meaning that I'm cleared for travel.

Not sure why the checkin person failed to see no ESTA as that number had to be added to the reservation. How was the boarding passes.even issued without the linked ESTA?