Why doesn't seeing things and hearing things prove that they are there?
In many cases it can. You noted that eyewitness testimony is admissible in a court. However, just because a witness in court claims to have seen and heard something doesn't
ipso facto prove that it actually happened. The testimony is appropriately questioned in an adversarial system that follows rules for fairly testing the claim. Then ultimately a trier of fact decides whether that claim is credible.
That mimics what happens informally when someone claims to have seen and heard something. The credibility of the witness, the circumstances surrounding the claim, and the
prima facie nature of the claim all go into the calculus of determining whether a useful standard of proof is met.
"My mother appeared to me in a dream and told me something important," is a fundamentally different kind of claim than, "My mother appeared to me in my living room and told me something important." If the claim is that each appearance is a literal attempt to communicate, our knowledge of the nature of dreams in general informs our relative evaluation of the
prima facie plausibility of each claim. Skeptics are fond of saying that extraordinary claims demand extraordinary proof. That's debatable, but in this case the claim
per se is the proffered proof. The question is not whether the claimant can prove his claim in either case with additional evidence, but whether we should take the eyewitness claims themselves as proof of the matter of the mother's appearance and message.
Someone steeped in Christian belief may not even bat an eye to hear that some figure appeared in a dream and delivered an important message. Within that belief tradition, that's a common axiom. Outside the belief tradition, and especially in science, it's an extraordinary, unproven method. Dreaming as a communication conduit on par with an in-person conversation has no scientific support and considerable scientific counter indication. The claim is improbable if not impossible on its face and therefore offers no evidentiary value.
Your particular line of questioning touches upon mental illness, specifically schizophrenia, in which the sufferer may claim sensory data that we can know objectively did not occur. We know that an unhealthy, off-nominal brain can have imaginative mental states that create recollection of sensory experiences that did not occur (hallucination). We know that a healthy, perfectly functioning brain can have imaginative states that create a recollection of sensory experiences that did not occur (dreaming). In one case there is metacognition to supply a lucidity control. Dreamers tend to know they were dreaming. Hallucinators can be trained to recognize when they are hallucinating but do not start out with this skill.
Experiment has shown that even a nominally healthy, nominally lucid, nominally engaged brain does not always form an accurate sensory recollection of events that are simultaneously captured via objective means. This means there is no limiting factor on the error in seeing and hearing. Nor is there any control in the brain to determine that this has happened—no lucidity feedback. Because even under the best possible circumstances we cannot trust eyewitness testimony, claims to have seen and heard something are not by themselves adequate proof that it happened that way.
While the Christian belief tradition provides a ready hypothesis to explain NDEs, the observation
per se is not proof of the hypothesis. Hypothesis testing in religion is not especially rigorous, nor desires to be—faith, etc. Similarly, the secular tradition also provides a ready hypothesis by noting that the circumstances of the NDE very closely aligns with other imaginative, semi-lucid mental states. Because science applies a far more rigorous method of hypothesis testing, it doesn't claim to explain all outstanding details. That is a difference in standard of proof, not in the strength of the hypothesis. The advantage science has is in the structural avoidance of circularity. In attempting to explain NDEs, science relies upon principles previously proven by other methods.
The notion that an NDE provides evidence in the form of something seen and heard does great equivocal violence to the concepts of seeing and hearing. The circumstances under which non-veridical NDEs occur thoroughly precludes seeing and hearing in the way those terms normally apply to evaluating eyewitness testimony. It conflates the concepts of seeing and imagining. Hence it's improper to expect them to be considered equivalent eyewitness evidence of the existence of the thing seen or imagined.