I can't believe that you are actually saying that. They compete in that Disney has completely taken over that whole side of the street, including El Capitan; and Grauman's will never therefore again host a Disney premiere (as it famously did for Mary Poppins and several other classics). They both show movies and host big premieres, but one is Disney and the other is never Disney now. They are just as much in competition, from the theatre and tourist attraction perspective, as Disney and Universal. Yes, you can buy a combo ticket on L.A. tours (unlike in Florida), but they will never actively promote each other.The El Capitan and Grauman's aren't in competition with each other. The El Cap plays solely Disney films, with one theater, so one Disney movie at a time. Grauman's is a regular theater, two totally different motives.
In fact, even a few years ago, when Kimmel first started, you could see a part of the Grauman's Chinese in his opening credits, but just barely. The only reason was that Mickey Mouse's star was in front of it on the sidewalk.
I am not saying that it is a big deal. Just that I think that they do not have (or want) to promote the Chinese Theatre any more than they do Universal. They probably would have to pay royalties to a property that they do not own.