The deal is growth vs changing shape, the radial will change shape and the sidewalls/shoulders will stand up but if you taped it the overall diameter would have changed almost nothing. A slick by contrast will grow roughly 1-1.5 inches or so depending on the design etc. The problem is if you take out the technical jargon the radial is still basically getting bigger in certain areas like you suspect and if there is not enough clearance it can rub especially at the shoulders of the tire.
