If a control system's output is uncorrelated with what it is controlling, then it's not really controlling it.
Also, to see a clear linear correlation, you might need to know what you're looking for. A controller output may look uncorrelated with the target system state, until you realize it's a I controller, so you need to look for correlation between controller's output and the integral of the system's state.
Let’s say there is a system that controls whether a second system receives input from random source A or random source B, and there is some difference in the distribution of values generated from each source.
Starting without knowing the relationship between the two systems, is there a good general approach that could pick out what it is doing?
Replace the system with one that chooses A only; replace it with one that chooses B only; analyze distributions; analyze original system.
Or replace A with known non-random source, and then B, and analyze.
I'm not sure what the argument is though. If something has an effect, there will be a correlation. Measuring the correlation may require modifying the system. We've known that measurement unavoidably affects the system under measurement for some time now. It's just a matter of degree.
That correlation ought to be pretty quick to spot once you remove or turn off the control system. The control system in your example sounds binary, so on / off sounds like the only input (to the control system) to correlate.
Also, to see a clear linear correlation, you might need to know what you're looking for. A controller output may look uncorrelated with the target system state, until you realize it's a I controller, so you need to look for correlation between controller's output and the integral of the system's state.