Okay, I can chime in here, having lived in the tropics on a boat with only an ice box. I will say that the first thing that crossed our minds when making landfall was "Ice!"
While a 'fridge is nice, it does make for more weight
, complication, and expense. An icebox is dead simple.
First of all, I would consider adding or improving the insulation around the box. I wouldnt be surprised to see nothing but a few wisps of fiberglass insulation, and you can do MUCH better now with materials like extruded polystyrene (your basic blue or pink board insulation in 4' x 8' sheets from Home Depot). The more the merrier. If you can get three or four inches all the way around, you'll be in great shape. Stagger the corner joints and tape things together. A layer of foil in there somewhere is good also.
Another thing that helps is to have something like a space blanket, or a piece of Reflectix that drapes over the front of the food, so that when you open the door all the cold doesn't immediately "sink" out of the door. In fact, boat iceboxes have a top opening lid for this very reason. Also, plan your "moves" in your mind before you ever open the door. "Okay, I'm going to reach in, over to the left, and get the cheese." Etc. With good insulation one can go ten days or so with a couple of good ice blocks, if you're careful about opening it (at least in a top loader).
The next component is the ice. It's getting very hard to find good block ice in the U.S. anymore. It's usually either cubes or what I call "fake blocks," which are blocks made of cubes pressed together. They're useless. So you are right to make your own. The bigger they are and the colder you can freeze them, the longer they will last. One larger block is better than two smaller. A dishpan is a handy size, and I've asked shopkeepers and restaurant folks if they would freeze a block in one for me overnight, at times. Closed containers like milk jugs are less messy, but smaller.
Also, the ice should not sit in water (I imagine your ice box drains but it may not). Rig up a drain, and make sure there is a water-trapping loop in the drain along the way. This is important to keep the cold from rushing out the drain through only air. Depending on how this is set up, you may be able to use the meltwater to cool, or to pre-cool beverages. You can also start out with much of your food cold or frozen, if you are leaving from home.
If you find yourself going in and out of the box a lot for beverages, or certain snacks, you might consider a separate small free-standing cooler, just for those items. Maybe one you can collapse after it's done.
Then there is (as Gina mentioned) the fact that people in the U.S. tend to refrigerate a lot of items that really don't need it. It's just become habit. For example, even mayo will keep fine unrefrigerated, as long as you never put a contaminated knife in it. The squeeze tubes make this easy. Also, I've had eggs last for months even in the tropics. You just need to keep air from migrating through the shell. The easiest way is to simply turn them over (the whole carton) once every day or two (there are other methods also). Eggs that have never been refrigerated or washed keep vastly better; I used to ask a farmer to set some aside for me that way. I could go on and on about the foods, but I'll stop unless you have specific questions
I hope this helps - just give a holler if there's anything I can clarify.