A fraction of revenue from popular pages should fund a manual fact checking process that either temporarily grants or denies that page a trusted status.
People don't care about "fact checking". The NYT, or the Wall Street Journal's newsrooms simply don't make factual errors, except occasionally spelling someone's name wrong, Nobody cares.
Your counter example uses a trusted source of info. The people who actively read those sources likely DO trust them. Facebook lets anyone post anything (to the extent that fake news is a huge problem), and reaches a much larger audience.