(Bioinformatician here). Although I think bench work is the most obvious route and the most likely way these problems will be solved, there are in principle some computational ways they could be addressed.

If we had good computational models of how perturbations would affect transcription networks, for example, we could predict these "side pathways" that so often occur in humans but not in mice.

But you're right, the reality is complex, and you have do deal not only with transcription, but translation, post-translational modification, non-coding RNAs, the list goes on... And most current models of this type don't delve into 3D simulation. Some people are working on whole-cell modeling but that's in its infancy.

Ultimately I believe the breakthroughs will come fastest if we can "close the feedback loop" by automating a lot of bench biology, and then have computers both generate and test hypotheses.