If you're generating enough heat from entropy alone to melt your entire space craft (which is thermodynamically impossible because of equilibrium conditions) then you're generating enough heat to run a heat exchanger steam generator to the tune of a few dozen megawatts of energy.
Yeah, but that's the problem with your theory. I have no doubt that starships get every last possible amount of use out of waste heat, but what do you do with those few dozen megawatts of energy? You run stuff with it. Stuff like Troi's curling iron, Washing machines for Wesley's sweater, Spock's plug in Vulcan fiery idol thingy. Each of these take that power and do something with it. That something always produces waste heat. Even refrigerators cheat by removing heat from one area and putting it into another (your kitchen) Entropy says you will always have more of that heat than you need, so you have to get rid of some of it.
But again, this only applies to a closed system where the only thing you can use is the heat already in the system. "Entropy" is, by definition, useless heat that you do not have enough energy to reclaim (because moving that heat around requires a certain amount of energy too). Starships have fuel supplies which they get from somewhere else. This means they are OPEN systems, and entropy is no longer a problem.
Back to the water tower analogy: you don't need to dump the water into the sewer, you just need your pumps to send the leaking water back into the tower at least as fast as it leaks out again. Sometimes you might want more water in the building, sometimes less, but you can control the amount of leaking water in your apartment just by adjusting the pump speed and making sure they have enough energy to function.
Heat sinks, no matter how efficient, are just a way of putting off the problem. You have to radiate off the heat!
Only in a totally closed system. Sooner or later, though, you would run out of even useable heat and you would have to open the system anyway to get more energy. If the system is already open, this is not a problem.
Closed loops, depending on the situation, do one of 2 things. They either wither and die, or run amok and explode.
Which, again, is why it's a good thing starships are not closed loops. The bigger flaw in your objection is that 1) starships have to make use of a fuel supply they apparently collect from the ramscoops or other means and 2) no matter how slow heat radiation from the skin is, NO starship hull is a perfect insulator and it will radiate some excess heat whether you want it to or not.