The latter seems like an aspect of misogyny though?
Like. A First Lady wielding power 'through her husband' is just a First Lady wielding power as First Lady, since that is a position defined by who she is married to. Which, y'know, sure sounds like saying that the First Lady should hold no power at all and be a purely decorative position. Insert the standard discourse about the Patriarchy.
Then again I'm sure there's plenty of Hillary-specific reasons (and Clinton-specific reasons) conservatives dislike her.