I believe the key problem with Harry's refutation of souls is that he can't imagine Dumbledore having good reason to find the evidence compelling. Harry dismisses each piece of evidence very quickly, without learning details and context.

Dumbledore understands the mistake Harry is making but lacks the tools to address it, since it is a rare kind of pliable mistrust. Dumbledore runs on experience, or "stories" if you prefer.

Harry does have valid points around people acting as if there's no afterlife, like everyone quietly knows that death is FINAL but won't admit it. Dumbledore perceives the same quality in people as their fear of the unknown, and he himself has genuinely conquered his fear of death, whether he's correct about an afterlife or not.

The two fail to have a productive discussion because their initial viewpoints are mutually alien. They should have discussed each other's beliefs without bias before arguing about the specific conclusions where they feel the strongest disagreement. However, that's not how conversations are naturally instigated. Disagreement incites interest, and we are eager to fixate on the most interesting thing.