The only thing that stands out as controversial about her books was:
> In the book [Expecting Better, 2013], Oster argues against the general rule of thumb to avoid alcohol consumption while pregnant, contends that there is no evidence that (low) levels of alcohol consumption by pregnant women adversely affect their children. This claim, however, has drawn criticism from the National Organization on Fetal Alcohol Syndrome and others.
I don't have any strong opinion on the topic of the claim[0], but the way it is reported as having "drawn criticism from [Some National Organization Specializing in Arguing the Opposite of the Claim]" in my experience pattern-matches in favor of the claim.
So beyond that, what's exactly so controversial about her books?
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[0] - Like everyone, I've had "zero alcohol while pregnant or breastfeeding" message drilled into me for decades, and that's my prior, though I've personally heard an experienced gynecologist, widely respected in the region, giving pregnant women allowance for a small glass of champagne for New Year's Eve...
I've come to see "No alcohol while pregnant" as the better simple rule. The alternative is "No more than X drinks in a day, and no more then Y drinks over your pregnancy." There will be a lot of people failing to remember the numbers (especially after a drink), the numbers vary by body type, the effects vary a lot between pregnancies for no known reason, drinking impairs decision-making, and the bad outcomes harm somebody besides the mother.
Public health guidelines targeted at non-professionals need to be simple. Same idea behind telling somebody who just wants a retirement fund to park their money in a managed portfolio, then ease those investments into safer stuff close to retirement. You could totally explain things in more detail, giving them more options and better opportunities, but most people are likely to not understand enough to safely do it. (I probably explained it incorrectly, which would reinforce my point of the general public's ignorance.)
> Public health guidelines targeted at non-professionals need to be simple.
And that's how you lose their trust. Just telling me to do simple X when it's clear to me from my own experience that sometimes not-X is better for me will make me think your overly simplifying and that your advice is not to be trusted, no matter your credentials.