Modern science gives us proof that the ancients were right when they said pathos was the most powerful part of rhetoric.
Our emotions rule our brains. This isn't a bad thing, or a weakness. We need emotions to assign value to choices and weigh them. They're essential to making a decision.
Doctors have studied patients who've suffered construction accidents or strokes that precisely disabled the emotional center of their brain. They performed just as well on intelligence tests, and otherwise were uninjured. Yet these people they got divorced and lost their jobs and found it impossible to make good decisions, even on simple things like whether to have lunch on Friday with a childhood friend.
When these patients made bad decisions, like losing money on a stock deal, they'd repeat the mistake, because it was logical to them to make that same decision and there was no emotional cost to failing the first time.
Why are emotions so critical?
Because facts and arguments mean nothing if you have no way of weighing them against each other.
It's not wrong to appeal to emotions.
When you tell a story about injustice, it arouses anger and resolve in your audience to change things and make it right.
When a speaker appeals to a crowd's sense of patriotism by saying that he sees it as his duty to vote during every election, because his grandfather fought and died to protect freedom and democracy in World War II, that's an emotional appeal -- but it's not cheating or manipulation. It's about one man's values and how he translate into a bigger picture.
Pathos is important because persuasion, in the end, comes down to inspiring people to do something. You can't inspire somebody who feels nothing about an issue.