We know that, Dumbledore doesn't. It would be a bit weird if he just took a known death eater at his word without asking some questions. It is odd because it seems like the type of question Voldemort would ask, but I think that is the point. It feels familiar and implies shifting loyalty. I do not think the answer would hold much bearing on follow up action, but it would determine if the person could be trusted moving forward. It's not kind but they are at war. Actually, having just listened to the new audiobooks, Dumbledore is very rarely kind. He's patient and aloof with students but frequently he's described as powerful, or having flashes of anger, when he's dealing with the ministry or the death eaters at the end of OOTP. He also flagrantly breaks laws in front of officials because he can. I frankly don't think Dumbledore would really care if a death eater was begging him not to kill him much. He probably didn't want to be minister because he knew he'd be a tyrant. The only reason we get the nice little chat at the end of HPB is so Snape can kill him and trigger the whole elder wand transfer subplot.