Most of the time I would protest removing auto traffic from existing streets because even when there is not much pedestrian traffic, cars parallel parked on narrow streets (the former coolest street in Cleveland, East 4th before it was turned into a pedestrian mall that is not frequented as yet) provide a buffer to the street, as well as add density/clutter to the urban landscape. Take away the street, parking meters, traffic lights and street signs, and the human scale is ruined in an attempt to create a false environment to attract tourists.
But, after all that, removing the streets through Public Square would help create a viable civic space. What does the city do on 4th of July weekend when the orchestra plays on Public Square, or set the world record for a balloon launch (back in the mid-80's) to pack in 100,000 people? They close down Ontario and Superior through the Square.
I say terra-form (SimCity 2000 term) the Square up 15 or 20 feet above current street level and get a contemporary architect/landscape architect to redesign the new level. Think of the new vantage points available down the canyons of Euclid, Superior to the east, and hopefully soon to the west as well, with a series of changing topography.