All up-posts hit it right on. It is the property that sets the policy. One can certainly buy a room with breakfast options (served in a restaurant).

That said, I often do a little home work prior to booking. e.g: use info on this board and personal experience would steer me toward one hotel and not others (if my criteria is about free and better breakfast). Just an example. In the NYC market, there are several properties to choose. Marriott Downtown (formerly aka Financial Center) often offers elites breakfast in the restaurant (even though the CL is open). If the rates are ok with me, then it is my choice. If I found a better rates uptown in a hotel that I know does not offer breakfast in a restaurant (BIG if), I would book it and then eat breakfast out!

Often than not, choosing a hotel is an art of "trade-off"