The NFL is primarily sponsored through network agreements, which themselves are primarily sponsored through advertisers. Is the "core business" of the NFL, then, advertising?

Of course it isn't. Such an argument is specious to the point of uselessness, and only appears in the form of rhetoric. The NFL's core business is making an entertaining sports product that people like watching, and then they find ways to monetize from that.

Apple operates and promotes its own ad network, and its lack of success in that market doesn't somehow separate it from that reality. No one would claim that Apple's core business was iAds.

IBM's core business is monetization, by definition. Making services that enterprises like to use is what they do to feed that core business, not vice versa.

Walmart's core business is monetization, by definition. Selling goods that people like or need is what they do to feed that core business, not vice versa.

Toyota's core business is monetization, by definition. Selling cars that people like or need is what they do to feed that core business, not vice versa.

The local florist's core business is monetization, by definition. Selling flowers that people like is what they do to feed that core business, not vice versa.

Every for-profit's core business is monetization, by definition. Performing a large variety of things that make money is what they do to feed that core business, not vice versa.