The only suggestion that oil cools everything, is by statements about it's irrelevance in doing so. Even in low output engines, oil has a critical role in keeping the temperature of various components of the valve train at tolerable levels. Valve springs, stems, tips, followers, etc all depend on oil to carry away heat. High output engines use oil for direct cooling of pistons and cams as well.
Engine oil cools things two ways. One is by carrying away heat in its circulation. The other is by conducting heat rapidly from one surface to another such as a valve-stem to guide, or cam-follower to block. Thinking the temperature increase of the top end after shut down is entirely because it "has nowhere else to go but up" is a bit green.