Because if the government is the USA, who is really going to go to war with them over this?

You're not wrong, information/propaganda warfare is aggressive. Sometimes it makes more strategic sense then open warfare, and sometimes states can get away with it when they can't get away with open warfare.

How do governments airdrop supplies into war-torn countries today? I’ll admit I’m no expert but I see headlines all the time of neutral nations sending supplies to other countries who are fighting a war and that neutral nation isn’t getting dragged into the fight.

Governments don't airdrop supplies into war-torn countries today, excepting designated UN or MSF refugee camps, and they do so with the knowledge of one or both sides of the conflict so that the supply planes don't get shot down.

I would very much like to see a headline of a neutral nation airdropping supplies into a country engaged in war without getting dragged into the conflict. I'm not aware of that having occurred in the past 4 decades.