So what would the conclusions be in the two-test and large-test case? My assumption would be that if one of the two tests would fail, then the correlation in a large test would also have been so low as to mean no effect?
The actual effect depends on your data and how noisy it is, and if that correlates to external factors, etc.
You need to know the characteristics of your data then you can decide on your desired confidence and only then pick a test designed to give you that based on your data.