The symptoms fit an overheating problem. I note you have already checked the cooling fins. Did you check the fins right across the head, especially underneath the top of the cowl (the most important part)? Are the fins on the flywheel fan present and clear? Of course a severe head gasket leak could be the problem too, but such leaks have to have a cause. Unless someone has loosened one or more of the cylinder head bolts at some time, blown headgaskets on side valve Briggs engines are more often a symptom of overheating, than a cause of it.
