>Fox News and watching the Simpsons on Fox is a more direct analogy. Any problem with the branding there?

How? Nobody gets confused between a cartoon and a news program. They are completely differently offerings, branded completely differently.

Buzzfeed, on the other hand, built up a reputation over years for pumping out lowest-common-denominator trash journalism, then later set up a serious news website under the same brand.

It would be more like if Fox News bought out the BBC and called it "Fox News UK".