That has to be by design.

They're two separate bridges built decades apart. They put a bike lane on the new eastern span despite it not being very useful at the moment so that whenever a new western span is built it won't run into the exact same issue (and there's some ideas being thrown around for adding a bike lane to the current bridge, but they're all hideously expensive).

Thank you for providing a least-hypothesis that is even more probable than "ordinary racism/classism". :)

there isn't a design, which is the problem. it's two bridges and people in the Bay Area are lucky to have one of them in-the-can. There's been a lot of argument over the design of the western spans; I despair of ever seeing them built before one of the faults underlying the area renders the argument moot.