Getting back to the Rownham Ferry, the Ashton Avenue Bridge is one of the reasons it was closed. On the excellent Bristol City Docks website you can find a photo called "Rownham Ferry c1906 with Ashton Avenue Bridge nearly built.".
As you'll see in that photo, at low tide the Rownham Ferry was basically just a bridge made out of boats, so it probably still seemed a quick shortcut compared to walking all the way to Ashton Avenue Bridge. I imagine the foot traffic gradually migrated to the new bridge as the ferry was a more daunting prospect at high tide, from what I can gather. Also, the combined road/rail bridge would have allowed people to get the train across the river, rather than stopping at Clifton Bridge Station and using the ferry, I think.