I'm not an expert on this stuff at all, but are the toy problems basically formulations of Simpson's paradox? My sense was that if you have a set of a data, where if you cut it into two slices and each slice suggests the same (identical) conclusion, but then if you look at the data overall it suggests a different conclusion, then how do you know which approach is correct? And it seems the only way to know that is to use your judgment about causal factors that generated the data, and the do-calculus helps make that explicit. For instance, when thinking about gender ratios and grad programs, it's more probable that the choice of what grad program to apply to is affected by what gender you are, but it's less probable that the choice of your grad program affects your gender.

Well I just reviewed section 3 of page 584 of that pdf that was shared and it seems different, so I probably have a long way to go before I understand this stuff.