I used to have a '78 H25; after much crawling around underneath the cockpit to plumb a larger water tank, I figured out that the ice box was designed to drain direct into the bilge, and let the bilge pump take care of drainage. Because the ice box drain sits below the waterline, there isn't really any way to do a direct-overboard discharge without routing the house to an above-the-waterline fitting and use some kind of pump.
Alternatively, you could install a sump under the floorboards, but there isn't a lot of room for that...
You are correct, though, in that a direct-drain setup will allow cold air to drain directly into the bilge. This can be stopped by using a cork in the drain line, and only opening the drain when you want to drain the box.
On our boat, we left the solid floor intact, and just used a small hand-pump to drain the icebox into the sink. A few swipes with a sponge afterward, and the box is dry.