What I'd do: Add a 3/8 tee to the hot side and connect the dishwasher to a hot water source, as God intended. Add a valve to the old dishwasher cold water supply and connect the fridge to that.

There's no harm in leaving the dishwasher connected as it is and putting a tee on the cold side. However, a dishwasher issue will result in having to shut off the water. Only costs a few bucks to add a valve there.