My trailer has a built-in ice box, originally installed in 1948. It was designed to hold a 20# block of ice in a tray on top and about a case of beer worth of goods in the bottom. A tube drains the water from the tray out through the floor of the trailer.
The door is at an angle and doesn't open all the way to the bottom of the box, so it doesn't let out the cold every time you open it. The biggest issue is block ice. It isn't widely available anymore and even then, seldom in 20# blocks.
Because the insulation is 65 years old, it no longer holds the cold well enough to be food safe, so I use it for beer only, but it's a valid concept, and if you plan around using cube ice and good, modern insulation, I think it could be done.
I'd probably use 2" rigid styrofoam insulation around a welded stainless inner box (to prevent rust) with an over-center latch to hold the door closed against a good seal. Draining through the box is a nice touch, but implementing it could be a real PITA. Maybe a drain in the back that doesn't go through the food compartment would work.
You can see the door on the left side in this photo.

<Chas>
