I don't think the proof is your passport. And if an airline allows you to fly into a country and that country subsequently refuses you entry because of no visa, the airline has to return you at their expense. How is it cost cutting, and who by?
I flew via the US in the winter to NewZealand and though only transiting, not entering America, had to purchase a visa online and was advised to take a hard copy of this with me. I was asked about this visa at check in at Heathrow.