I tend to drive to Aldershot, from near Gage Park. When the westbound York was ripped up it was horrible (though necessary) and I often took my chances over the Skyway in the morning... or if I really felt like gambling it was Eastport across the lift bridge.
But I think it will take some time for it to be less convenient to drive to Burlington. The people who won't, at the start, will be those who primarily take transit and those who would rather not deal with the traffic stress. Days when there is an incident on the bridge will be hell for all commuters.