The problem with admissions to UC Berkeley for homeschoolers is the same as the problem for everyone else: sheer numbers. Berkeley got over 101,000 applications last year. California is so large that there are almost as many separate high schools as there are spots in the entering class at Berkeley. That means that the average high school gets about one of its students into Berkeley each year.
Add in the fact that the UCs have made a deliberate decision to emphasize grades and course rigor and deemphasize standardized test scores. They want the valedictorian from every school in the state to have a shot at a Berkeley degree.
Of course, there are six UCs ranked in the top 50 (Berkeley, UCLA, UCSD, UCSB, UCDavis, UCIrvine) so a great education is available to all the top California students - just not necessarily a spot at Berkeley.