I am enraged that I have to pay for such a horrible rider experience, that I have to subject my small children to the horrific conditions of the BART elevators, their filthy cloth seats and their terrible customer service.

You don't have to. You choose to. Assuming this enrages you, you should probably stop doing it. There are other transportation options in the Bay Area besides BART, including busses (many of which can transport bikes), rental vehicles (including ad hoc rentals like ZipCar), and owning your own vehicle.