Because Google can only index what they can see. As noted in the article, the Googlebot only gets to see the first few paragraphs of WSJ articles, so those pages are less likely to rank on searches.

Indexing more of the content (which would be possible by providing the full content to web crawlers) seems to violate Google's cloaking guideline as well: https://support.google.com/webmasters/answer/66355

It's not excluded. They present less content to Google users and are ranked based on the actual presented content.

If News Corp wants search ads for it's paid service, it can by then like anyone else instead of expecting Google to treat them specially, which is what they are really asking for.