The comparison between Caltech and the Ivies is a bit unfair. Caltech is in California, which has more Asians, and people tend to attend schools near home. Adding in Stanford would be illustrative.
The other problem is that this article automatically assumes that private schools are "elite", and public schools aren't.
That's true for certain fields-law and finance, for example, are dominated by the ivy league schools. STEM, on the other hand, is a much more even distribution. Many of the engineering programs at most of the Big 10 state schools are considered to be some of the best educations you can get, and the UC system is full of very good programs. UIUC's CS program is considered one of the best in the nation, as are Purdue's ME and aeronautical engineering programs.
Stanford is 41% from California. Caltech is 32% from California. Thus, it looks like the California factor has little if anything to do with the Caltech vs. the Ivies Asian data.