> Mathematically, as lim temperature->0, the distribution gets spikier and spikier, the most likely sample goes to almost-but-not-quite infinity and the rest go to almost-but-not-quite 0.

That's not how limits work. As the temperature goes to 0, the rest goes to 0. That's it. The "almost-but-not-quite" is part of the "goes to".

Let's say f(x) = 3x+1. It's a continuous function. If we let x go to 10, f(x) goes to 31. Not "almost-but-not-quite 31". No, to 31. (If you don't have a continuous function then it's the same argument, but less intuitive to illustrate.)