My old CRC Handbook of Chemistry and Physics says for pure air-free water at 1 atm absolute pressure and 0C temperature, the Heat Capacity is 4217.7 J per kg per degree Celsius, and the Heat of Fusion is 79.72 cal per gram, or 333,548 J per kg (if we use 1 cal = 4.184 J to define the calorie). So, to cool a kg of water from 1C to 0C you have to remove 4218 J of heat energy. But then to convert it to a kg of solid ice still at 0C, you have to remove a further 333,550 J of heat energy!! That's almost 80 times as much!! This is why flowing water seems never to freeze. The rate of heat removal from the cold water is slow enough that the water you are watching does not loose heat fast enough to freeze before it is gone from view.
Sure, there IS frozen ice near the edge of the stream (or hole in the ice). At its surface is a dynamic equilibrium. For every gram of liquid water that freezes to add to the ice block, enough heat has been removed from it to melt another gram of ice back into water. The only way for there to be a net increase in the ice mass that we happen to be watching is for all that heat to be removed from the water entirely, say by currents of air below 0C flowing over it. And in fact, in a sustained cold period the open water area slowly gets smaller as more ice builds up around the edges. But if there's any other heat source (say, sunshine with its infrared components), the input of heat may outweigh the removal by moving air, and the ice edge melts back a bit. The whole thing is a constantly-changing equilibrium disturbed by short-term external influences (air and sunshine being external to the water / ice system).