Thanks to BNT83 for posting the TCDS for the O-320.
The pertinent information, which he circled in red, is that Lycoming recommends you not take off with less than 6 quarts of oil in your O-320. On the other hand, Lycoming divulges that the engine will safely operate on two quarts. Why the difference?
It ain't cooling. It is oil consumption.
An extended range (52? gallon capacity) C-172 or a 160 horse Cherokee (50 gallons) has sufficient fuel to fly ~6 hours at ~65% power (roughly 7.5 GPH). Lycoming's 6 quart minimum for the O-320 prevents you from running low (or out) of oil before you run out of gas on your 6 hour flight. (Why anyone would want to spend 6 hours in a 172 is another issue entirely.)
Lycoming's published limit for maximum oil consumption is HP x .006 x 4 /7.4 = .52 Quarts/Hr for a 160 HP engine. So if you launch with 6 quarts of oil, consume oil at the maximum allowable rate (.52 QPH), and fly for 6 hours, you'll be down to just under 3 quarts of oil at the end of your flight, which is comfortably above the 2 quart "safe operating" minimum. This means your endurance is limited by fuel, rather than oil. That's what Lycoming wants and is how they determine the minimum "takeoff" oil quantity. Throw ferry tanks in the back and all bets are off!
To the idea that the sump rejects heat.
It does. No doubt. But does that rate change in a meaningful way with the addition of a quart of oil? I don't think so. A quart of oil is ~58 cubic inches. I'm guessing a typical 4 cylinder Lycoming sump has a footprint that is 12" x 12" or 144 square inches. The IO-540 sump in my garage has a cross section slightly larger - about 13.5" x 12" or 162 square inches. Where this is going is a quart of oil increases the oil level in the 4 cylinder's sump by 58 in^3/144 in^2 = 0.4 in. That means the additional quart of oil inside the sump touches a band 0.4" high x the 48" perimeter of the sump. Roughly 19 in^2. How much additional heat transfer is 19 in^2 of internal surface contact going to create? Not much, since the sump is already "at temp" due to the great heat conductive properties of aluminum. Also, the area on the outside of the sump exposed to cooling air won't change, and that cooling air that impinges the sump ain't exactly cool in the first place, because it has passed through the cylinder fins and around the exhaust headers. So, a quart of oil won't do anything meaningful. (Note, I used the word meaningful in my original post disputing the idea that oil quantity changes oil temperature.)