...people who question the veracity of the study, or they try to downplay it, or muddle the issue to deflect the import of what the study is actually saying.
I always question the veracity of any study. As a student of psychology, I especially take interest. The links provided by the OP left me with more questions than answers. I did some digging and found another article on the study that provides more information.
Brock Talon - The question you ask is answered in the article I provided. It is white persons who prescribe to "extreme liberalism" as political ideology that have found to suffer from mental illness at a higher rate. As Simon commented, that sounds more like Leftist ideology. Liberalism and Leftism tend to be interchanged as terms presently. They are not.
In reading the OP links I had a hypothesis that 1) the mental illness diagnosis would be neuroticism and 2) that the link between religious ideology and the lower percentage of reported cases of mental illness with persons with conservative political ideology would be mentioned. Sure enough, the study showed that.
Frederick Nietzsche correctly predicted by in the 1880s that rationalism would "kill" god and lead to one of two pathologies for effected persons - nihilism or utopianism through totalitarian government. Renowned psychologist/psychoanalyst Dr. Viktor Frankl calls these neuroticisms as the "existential vacuum." Neuroticism is a low level mental illness that is curable and does not require medication.