I have yet to meet the person who takes that position consistently, and only believes that which he has personally verified. Anyone who only adopts this view in response to selected issues is motte-and-baileying those issues.

Note that my response here does not rule out being skeptical over particular claims for definite reasons, only the blanket use of "Unless you're going through _all_ the source material yourself..." (your emphasis on 'all'.)

And why stop at source material? We know there is a replication crisis in science (the open discussion of which has, somewhat ironically, encouraged a wave of blanket dismissal of science), so surely this level of skepticism can only be assuaged by personally verifying all the original sources?