Even though refusing to do business with them will cause some of their family to die of starvation or lack of medicine?

Yes. It's the same reason I don't shop at Hobby Lobby despite a boycott potentially causing innocent workers to lose their livelihood.

They can find new jobs, the people you've killed for disagreeing with your foreign policy are dead forever.

I've found myself in an awkward position defending sanctions I don't really agree with anyway. I'm not sure these differences of opinion are reconcilable. Some people think the appropriate response to death threats is to ignore them, and others like me think the appropriate response is a cruise missile.

You realize that "Death to America" is not remotely a death threat? At worst it wants to see the fall of America and most Iranians say they just want the end of America's imperialist policies.

And you're against sanctions because they don't kill enough people.