Schools probably do do something

I think it's that yes, schools do something, but that something may have less to do with the things everyone usually thinks and hopes schools do, and more to do with "change how well they teach different students as a function of initial maturity in a way that locks them into better or worse future trajectories, based on an essentially random variable, namely their birthday."

Which is admittedly entirely consistent with your post as written, taken in isolation, but also very much at odds with the as-usually-understood spirit of what people around here who say school isn't that valuable are actually trying to say. Sorting people into buckets by birthday range and then using that to lock them into better or worse life paths, with little relationship to anything like the innate levels intelligence or conscientiousness that two different kids would have if each were evaluated at the same age rather than the same point in time, is technically schools "doing something." It is not schools doing what proponents of schooling, as it's currently done, want you to believe school does. It is not schools doing something socially or economically valuable.

That's exactly my point. 

Suppose we were doing chemistry instead of schooling (so that I didn't already know the answer).

I have two beakers, one labeled 6.01 and one labeled 6.99 and I then pour 12 units of "school" into each beaker.  Afterwards I discover that the beaker labeled 6.99 has produced 2x as much "oxford" as the beaker labeled 6.01.  The conclusion I would come to would not be that "school" is pretty good at producing "oxford".

Even more fun, I challenge you to predict the amount of "oxford" produced by adding 12 units of "home school" instead of "school".

Did they compare regions with different cut-off dates? For example, in NY it's in December, and in MA it's in September. Actually looks like it's 12/1 in NYC and 12/31 on Long Island, too. Those would help distinguish age vs sunlight or other seasonal effects on development.

I think "applying for Oxford" is doing more work in that sentence then you give it credit for - it isn't a random sample, it's already skewed to only include those 17 year olds for whom the relative age effect wasn't enough to stop them from performing well in school. One of my freshman year roommates at Harvard, and probably the smartest of us all academically, was 16, having skipped grades - obviously these people exist, and obviously people like parents, teachers, and guidance counselers are already steering students towards applying to schools that match their ability levels at the start of senior year of high school. 

The comparison I care about (and that age cohort studies are trying to approximate) for this question is how much more those 17 year olds would have learned, over the course of all their years of schooling, if they'd been a little older at the very beginning. To look at that, I'd want to see, for example, whether the proportion of students applying to Oxford, or becoming valedictorian of their high school class, are more likely to be older or younger.

I would strongly expect CEO-ship of a S&P 500 company to be causally downstream of a top-5 business school, a top-20 college, and the kind of high-status professional+social network you get from "more and better school".

I know that some schools in the US have historically had start dates in both January and September.