If you’re talking about getting warm (or room temperature) stuff cold, you want ice + water (if you add salt it’ll get colder faster), but if your beers are already cold you don’t need (or want) to transfer heat faster, you want to prevent the transfer of heat.
My impression of the original post was that we’re adding new ice to a cooler in which the old ice has melted, and the beers are already cold. The key to keeping stuff cold, and making the ice last longest, is to get rid of the melt water before adding new ice and draining the new melt water often. Air is a much better insulator than water is, so the ice will melt much faster if it’s sitting in water.
We did a two-week rafting trip a few years ago and we prepped our food coolers by putting them in a restaurant freezer overnight and freezing 4 inches of water in the bottom of each. We packed in all of the food in reverse order of use and mapped everything.
We had an advantage in that the chests were sitting in the bottom of a boat that was in 60 degree river water (coming off the Glen Canyon Dam), though the trip was in May/June and air temps exceeded 100 every day.
The coolers were drained of melt water twice a day and never opened more than a few inches (we had straps on them) or for longer than it took to pull the items we wanted. We ran out of ice in the vegetable cooler on/about day 13. The meat cooler (in which everything was frozen when packed) still had an inch-thick sheet of ice weighing 10-15 pounds at the pull-out on day 17.