Well first of all the authorities could have been a little more flexible (i mean wait for another hour for fax) esp. since they were dealing with kids. I am more appalled by the fact that a french national was allowed a visa if this is true. He should have been sent back definitely.
Thirdly, even though I have sympathy for parents but saying that "I am ashamed of being an Indian" is taking it too far. I am surprised that the person traveled without the correct passport and other documents. I am sure he would not do the same when traveling to US or other countries. The officers have as such taken an officially correct stand (even though it stands a little rude). Please dont make such statements publically or to your indian or foreign friends because an experience can be bad or good but it does not reflect the entire nation.
Tell your kids that you made a mistake rather than saying I am ashamed of being an indian etc.