Often the fastest way to get a boat to shore is the bulldozer method. Hooking the tip of your boat in the cockpit provides additional control and speeds the recover process. With more than one boater, someone can grab the downstream end with their hand and quickly push. This gets the boat into shore even faster. Setting a ferry angle helps to balance the forces and get the boat across faster. Often the fastest way to get a boat across a stream or into a safe eddy is towing with a rescue vest and tow tether. Make certain you know how to use the quick release feature. The tow tether is the most reliable way to run a rapid with a swamped boat as well. This can be necessary to avoid having the boat get pinned. Here is a video that cover this subject: Capsized Boat Techniques II.
In general, I wouldn't tow a boat full of water without some type of quick release approach. If you are not using a quick release tether and towing a boat with a painter, hold with your hand on the paddle. If things get too rough, you can easily let go of the rope. Some use a large loop made with webbing and sling over just one shoulder. That also works well and is easy to jettison.