Your suggestion to just minimize the residual correlations would unfortunately boil down to "analyze the data in the way that best hides any problems in the data" which isn't what anyone wants.

There are specific signal shapes/characteristics that would be producable by a gravitational wave. Anything that doesn't match that shape/doesn't have those characteristics by definition isn't from a gravity wave. Ignoring those characteristics and just saying "if it's correlated it's from a gravity wave" would be a highly incorrect way to analyze the data, which is why none of the scientists in the debate would advocate doing that.

I'm not advocating that any correlated signal should be considered a gravitational wave. I'm just saying that, if residual correlation is such a problem, then it would make more sense to create a combined signal by minimizing the residual correlation.

This discussion would then simply be about how well the combined signal matches a gravitational wave, which seems a lot more sensible.

I suspect they did that in part, but it's also critical to keep the signals distinct precisely because you expect the noise to be uncorrelated between the two signals so you're introducing an additional square root of two gaussian noise factor in the combination.

This sort of thing is much more about statistics and error propagation than one encounters in more traditional signal processing.