In the "Appeal to Pity" fallacy the example given is:
Jill: "He'd be a terrible coach for the team." Bill: "He had his heart set on the job, and it would break if he didn't get it." Jill: "I guess he'll do an adequate job."
The argument that Bill cares immensely about the job IS evidence that he would likely do it well. A coach who cares is more likely to work hard, be more invested in the outcomes of games, etc.
But the real problem, as you mention, is that people don't argue to logically prove something. People argue to convince someone else (often not even the person they're arguing with). If you can convince people you won the argument, you won the argument. It doesn't matter whether you appealed to base human emotion or made a logically/factually empty argument.