That "regional" government is disjointed and inefficient.
BART is run by Contra Costa, Alameda, and San Francisco counties; CalTrain is run by San Francisco, San Mateo, and Santa Clara counties. And so the two don't meet up, because there's not a single government for the Bay Area that can coordinate. (The new Transbay Terminal that joins them together is in part a state initiative.)
These kinds of ad hoc special purpose districts are an extremely inefficient way of running a regional government.
WRT the San Francisco vs. San Jose issue - the interests of these large cities in the Bay are largely aligned. They are relatively dense, are irritated by the obstructionism of smaller towns when it comes to housing and infrastructure development, and would like decision-making to be more fairly based on population.
(BTW, there are actually three large cities in the Bay Area, with approximately equal populations - San Francisco, San Jose, and Oakland.)