If closing the seacock on the lift works for you then that is the fix. I have read many post that state this is what they do before lifting their boat so they don't have to go though the aggravation of priming the system when they put it back in the water. If I am using the boat in fresh water application that may be an ok solution but not one that I would want to use. Why? Because I tend to forget from time to time and no matter how many notes I make and checks I make I tend to forget.
When I leave my boat in the water and plan to be away for a while I do close my Seacocks Air unit and engine. I place the cabin keys with key chain around the handle of the air marine seacock and engine keys/fob with key chain around the handle of the engine sea cock. If I didn't do that I would forget!!!!
In salt water cruising I don't like salt water and the critters that live in in hanging out in my cooling system when I am not using the boat for extended time. The last thing I want to do is trap that water while on a lift and allow this to get stagnate and the critters die in the cooling system. Let it drain out, or better yet flush it with fresh water and let that drain out.
If the Raw water cooling system is installed properly. You should be able to launch the boat, sea cock open, and the pump should have a lift capability of minimum of 3' self priming. This 3' starts at the water line. So from the bottom of the boat to the water line the 3' is not included. When the generator starts it should self prime and pump water. If it doesn't there is something wrong with the install. From the through hull to the sea water pump there should be a gradual rise. There should not be any loops or falls in the run. (reason? air pockets) The through hull being the lowest point then strainer then the water pump. if the strainer is higher than the pump it will not work correctly. PUMP IS ON TOP OF THE HILL AND THE HILL RUNS DOWN TO THE THROUGH HULL. If there are any hills in between the pump and through hull it may not self prime properly. Engine and Generator installation 101! If the pump is not self priming like a marine air raw water pump the pump must be installed below the water line. If it is not installed below the water line as per manufactures specifications the pump will not prime. Marine air unit seawater pump installation 101! One other element that must be met is the thru hull strainer, hoses and fittings must be (minimum) the same size as the inlet and outlet of the seawater pump for the raw water cooling system to function at full capacity.