One way to look at it is, as you point out, our estimates of a measure are more accurate when you take more measures.
But the major issue with more data is Bayes' Law. For decades, running tests and collecting data was expensive, so we only ran them on people who were sick or suspected of being so. For prostate cancer, this data would tell us the probability that one would have a high PSA measurement given that they have prostate cancer (p(high PSA|prostate cancer). E.g., we know that 70% of prostate cancer patients have high PSA.
But, as Bayes' Law points out, this does NOT tell us the probability that you have prostate cancer given that you come in for a normal checkup, and we find a high PSA (p(prostate cancer|high PSA)). E.g., we do NOT know that 70% of people with high PSA have prostate cancer (absent other symptoms).
In order to determine that, you need to know the probability of having a high PSA in general. And the point is, NOBODY KNOWS these population meaures yet. Collecting lots of data is a good way to establish these baselines, but it's not clear it will help with diagnoses at all.
Another real example: we've gotten better and better at detecting tiny tumors early. But once we started looking at the data, we noticed that a large fraction of tiny tumors simply disappear over time. So, when the tumor is small enough, it's not clear whether we should treat it with surgery/radiation/chemo (which have their own risks/downsides) or just wait-and-see.