I agree, the onus is definitely on the publisher to make sure the claims they make are statistically sound. However you're going to lose most of your casual audience if you focus on the statistics. You need to go to the source for that.
See reply to pak wrt to sample size.
"Support" does not mean "prove". How can you say it doesn't support the idea of emotional contagion when the studies indicate an effect for at least two emotions. Is there more work to be done? Definitely. But these are first steps.
EDIT
On skimming the links you gave, I find that the samples and the researchers' conclusion in each study do not constitute representative samples and empirical evidence for this “emotional contagion” theory, respectively. Both studies also focused on only 1-2 types of emotion (fear, anxiety), which cannot be extrapolated to all emotions.
APS published study: Sweat from 10 males + reactions from 36 females.
PLOS published study: Sweat from 49 people (28 males 21 females between 20–37, European origin) + reactions from 28 people (14 males and 14 females between 19–30 years).