This "broad conception" is pretty well documented as what Congress wanted at the time of passage of 14th Amendment. It's been considered "settled law" for ages. The only real reason it's come to SCOTUS is that a particular political faction wants it to, and the media gives that particular faction more credence, and more coverage. So there's two things here:

1. An artificially whipped-up "question".

2. Conservative bias in the media.