In hindsight I do think it's true that you want numerous experiments with possibly different controls with the exception of whatever is important for the effect

In hindsight I do think it's true that you want numerous experiments with possibly different controls with the exception of whatever is important for the effect. In that sense you want your experiment to "discover the right abstraction" of details that your principle works over. So I think what has to occur is a gradual "loosening up" of the controls from strict replication to weak replication, with both types of replication giving information about the effect you are testing: its validity and its generality, respectively.