That is Kevin Drum's take. Post 1:
In fact, the study showed fairly substantial improvements in the percentage of patients with depression, high blood pressure, high cholesterol, and high glycated hemoglobin levels (a marker of diabetes). The problem is that the sample size of the study was fairly small, so the results weren't statistically significant at the 95 percent level.
From a Bayesian perspective, the Oregon results should slightly increase our belief that access to Medicaid produces positive results for diabetes, cholesterol levels, and blood pressure maintenance. It shouldn't increase our belief much, but if you toss the positive point estimates into the stew of everything we already know, they add slightly to our prior belief that Medicaid is effective.
If this were the only medical study in all of history, then yes, a non-significant result should cause you to update as your quote says. In a world with thousands of studies yearly, you cannot do any such thing, because you're sure to bias yourself by paying attention to the slightly-positive results you like, and ignore the slightly-negative ones you dislike. (That's aside from the well-known publication bias where positive results are reported and negative ones aren't.) If the study had come out with a non-significant negative effect, would comrade Drum ...
Here's another installment of rationality quotes. The usual rules apply: