Why isn't dry CaCl₂ used to dry HCl gas instead of concentrated sulfuric acid?
There are several reasons:


Efficiency
When a stream of wet hydrogen chloride is run through a tube filled with coarse calcium chloride "rocks", interaction with the drying agent takes place at at its surface.
When the same wet $\ce{HCl}$ ist passed through sulfuric acid, the contact area is given by the surface area of the bubbles. Moreover, bubbles mean mixing. The trapped water is distributed over the drying agent.
Safety
Wet calcium chloride will clog, which might result in a pressure buildup in the drying tube.