This definitely possible. Due to the microscopic scale of bacteria and viruses, even very light forces(like those from pulling a tissue out of the box) will lead to those bacteria and/or viruses being tossed to the rest of the box.
However, a large portion of the infectious viruses in humans will struggle to survive on the dry, low nutrient substrate of the tissues and box for extended periods of time(especially given high absorbance), albeit the above comment show many that can(depending on the surface).
To answer why there is no better design, I think the simplest explanation is monetary incentive. There is little incentive to create something better both due to R&D cost and inevitable increase in product price, as well as a steep consumer education cost to let customers know, why this is better. However, if you have a good solution in mind that does not raise product cost beyond a few cents per unit, you should a decent shot(if you patent) to license it to the big players.