Harvard divides its credentials into "tiers", if you will. There are students who go through the competitive admissions process, and there's the Extension School which is basically open enrollment, even if you want to take courses for-credit. You only have to apply if you want to pursue a full Bachelors or Masters, and even then you're given the chance to prove yourself by doing good in a handful of courses.

I personally never thought I could experience Harvard before I found out that CS50 (and tons of other CS courses) could be taken for-credit and even lead to a degree.

Harvard also has online-only, non-credit programs like Harvard Business Online (formerly HBX) and HMX (Medical courses).

Harvard can, and does, strengthen its brand by opening itself to the public. There's a huge spectrum between being a totally-closed institution and being completely open enrollment.

Because it's Harvard's network and not the lectures that make the brand.