Look at it this way:
Would it be likely that -for example- on a zinc electrode, zinc ions dissolve and electrons are "generated" in the electrode, then the electrons hook up to something (what did you have in mind) to hitchhike through the salt bridge to the counter electrode, where they are recombine with copper ions to make metallic copper.
Now
1)You would have a 2-way flow of identical charge in the salt bridge (remember, law of neutrality, you do not have a net charge buildup in any of the compartments), is this a likely thing to happen?
2) even IF that flow would be possible, think of resistance, which can be HUGE in a path through electrolyte (many hundreds of ohms unless the electrolyte is concentrated and the salt bridge cross area huge and the pathway small)
3) that would mean copper would not be created at the electrode (which is useless then), but at the interface of salt bridge/copper solution because there is the new junction, (we know this does not happen)