In a written debate (as on DDO), facts are far more important than feelings and pathos. However, when we are talking about spoken debates, then pathos can be far more effective in persuading the judges/audience than facts.
But which is better? I mean, a good and ingenuous judge would appreciate actual fact than to fall for emotional arguments. Though if you feel the need for emotional rhetoric and language to sway the judges, that's fine, but don't make entire arguments or points based off of emotions.
Actually, the whole purpose of a fact is not be immutable to feelings. You can sit in front of a speeding train and feel like your not going to get hit, but that doesn't change the fact that you are sitting in from of a speeding train. That onjectively means that you are gonna get hit, regardless of the meaning you yourself give to it.
Your scenario makes very little sense to me. As does you saying facts have a purpose.
Are you actually trying to tell me that sitting in front of a speeding train means you will be hit by that speeding train? What if the tracks curve? What if I simply move before the train gets to me? What if the train is several miles away and slowing down?
This scenario you bring up confuses me because it's missing the point.
Let's say I give you a fact that "X people die every year for Y reason". What does that mean? What is our goal?
I would have actually found this one to be tough, if not for the description- Hurting someone's feelings in a debate is almost entirely irrelevant. If, however, this question had asked which was a better tactic, then at least for offline debates, appealing to people's actual feelings would likely prove far more helpful than using evidence. The art of persuasion at work. Also, a large part of why Trump can appeal to so many- Clinton has more solid policies on her side, and far more experience, but Trump wipes the floor with her for likability.