What happened with Vijay family was indeed very sad. Everyone comments were intresting. Extreeme in two sides. In my view we are mixing two issues.
Missing travel docs --Rejected. Very simple. Well, that is law, their might be exceptions like giving some time to get duplicate copies etc.. But I do not think they even accept copies of original, we should not forget how US implements law. Law is law and there is no eyes to it. There are still few good people in India, I am not sure if these immigration people among those good once.
Second issue. Corruption. I am sure each one of us got our own stories on this topic. No doubt our country is corrupted, needs change. So we might think that , when we are so much corrupted why can't we allow these kids as they belongs to Indian nationals. What if those immigration officers are among those rare good once and did their job.
In this case it looks like law is implemented blindly, it should be like this, but we all used to corruption when some one does right thing it bothers like this. there will be some exceptions for any law and looks like Vijay was not given any exceptions. What happened was very sad and bad luck.