This is also the same reason I do not normally consume water that has been in an RV holding tank. (certain situations notwithstanding).
The water in the 'fresh water' tank is anything but fresh...and especially if your RV has been parked in warm climates or on warm pavement for periods of time. Algae and mold and other unspeakables LOVE warm moist dark places!
If your ice chest has slots for a divider, you can buy or fabricate one and then you can keep the ice you will consume separate from all the other items in there: plastic bags, bottles, cans, wrappers, loose food particles, etc etc, (that you have handled with your grubby paws!) and have a reasonable expectation the ice in the separate compartment will be safe for consumption.
Daily draining of the meltwater is important also. Put fresh ice up on top of both sides if you wish, but don't consume the ice on the 'food' side of the cooler.
The world of icechests according to me...others are free to disagree and that's OK too. It's your health!