I'm not going to pretend to know the exact traffic patterns, but it seems to me this crossing would only exacerbate existing problems: on one side I-880 in Hayward is already chronically congested, and on the other this bridge would put downtown SF-bound cars onto US-101 and I-280, both of which are also chronically congested.
I would think the point of a new crossing would be to relieve traffic coming from the north and east over the Bay Bridge into downtown (and vice-versa), so what about this: a new bridge splitting off I-80 around Albany, heading SSE about 5 miles to the northern tip of Treasure Island, then SE into downtown, paralleling the Bay Bridge. It would touch down around Broadway and go into a tunnel which would lead under the Financial District and to a series of underground parking garages, plus access to the BART stations at Embarcadero and Montgomery. Limited access to get out beyond, including an additional toll, to discourage thru traffic (i.e., heading to the south or west parts of the city or down the Peninsula).
Typing it out makes it look more ridiculous than it was in my head, but just spitballin' here.