Instead, on April 12, 1962 the San Mateo County Board of Supervisors opted out, citing high costs and concerns over shoppers leaving their county for stores in Downtown San Francisco.

Keep in mind that Silicon Valley did not exist yet. There was no support for BART in San Mateo County-grassroots or otherwise. Many Peninsula civic leaders were also against it–including JR Clinton owner of the San Mateo Times (predecessor of San Mateo Daily Journal). Also opposing BART was David Bohannon the developer of the Hillsdale Shopping Mall, Tom Casey of the San Mateo Co Development Associations. They, and the San Mateo Times, published Op-eds arguing that BART was not needed; instead more freeway lanes was the proposed solution to congestion woes.

The neighboring Santa Clara County's obsession with building and publicly financing the Expressway System (Page Mill, Oregon, Central, Montague, San Tomas, Lawrence, Capitol) didn't help either.

You can read more about it in BART: The Dramatic History of the Bay Area Rapid Transit System by Michael C. Healy.

BART is a basket case of incompetence on its own. SF residents were largely willing to support an infill station at 30th and Mission, but BART stuck its thumbs up its ass and eventually built a half-billion dollar cable car to the Oakland airport (ripping up recent renovations to the street along the way).

San Mateo residents bought into BART in the 90s, but as they've learned public transit (BART especially) is a cost not profit center so the regional (SamTrans) service got gutted to fund BART (whose ridership numbers consistently lag predictions).