Edit: So yeah. Get the story AND get through the content. Once you're done reading. SPACEBAR.
This would only work if everyone read significantly faster than the dialogs take, which isn't always true. I know i've run into a few times during solo group story where no human dialog that must be read auto goes to the next page before I've finished reading it (super annoying btw).
However you're right, if I want to watch the stories and I'm doing a first time run in a hard mode I do my best to skim the dialog so I at least get the jist of the story. I recently did this with one of my first runs of a Republic-only flashpoint on my only republic 50.