Heat transfer considerations give us competing and conflicting objectives.
If our objective is to make the ice last as long as possible, then drain the box. The liquid makes the heat transfer from the sides of the box to the ice much more rapid, which leads to quicker melting of the ice.
If our objective is to cool the fish as quickly as possible, then don’t drain the box because the liquid makes the heat transfer from the fish to the ice much more rapid, hence more rapid cooling of the fish.
In no case will the ice be at a temperature lower than the melt water if you have melt water. If you pull the ice out of the freezer at -50 degrees F, it must warm up to the melting point before you will have any melt water. For fresh water, that would be 32 degrees F. If you salt the ice down, it will be substantially less than that.
The concern with the quality of the meat is a result of biological mechanisms completely unrelated to this discussion. Some fish will release enzymes that will cause their bodies to produce more heat if they swim into cold water. Blue Fin Tuna is an example of such a species. If the fish is still alive when you put it into the cooler, it will release these enzymes, which will degrade the quality of the meet.
The meat will degrade very rapidly if it is not quickly cooled as soon as the fish dies, hence the reason for the brine tanks to cool them very rapidly. A commercial brine tank is typically between 4-28 degrees F, depending on the design of the tank. A boat that has a very cold brine tank will also have a dry freezer at -50 degrees F to transfer the fish to after the brine tank. A boat with a warmer brine tank is probably a low dollar operation for short trips, and they may transfer the fish to simple drained ice storage.
All this concerns salt water fishermen. I’ve got no idea what the fresh water guys do; but they would not be concerned with the long duration storage times as the salt water fisherman who might be out for weeks or months. One friend who stays out until the freezer if full tells me that his shortest trip was three weeks and the longest a year, in his 35 years of operating that particular boat. A lot of short duration trip fishermen will simply leave them in the brine tank, but if they will get beat up from sloshing around if they leave them in too long.
As it turns out, osmosis is not a concern that causes any problems for the storage of fish.