One thing I didn’t explicitly mention in the post is that the average energy of the sample is a sufficient statistic for the temperature—it summarizes all the information from the sample relevant to the temperature. So in that sense, it is all we care about, and your intuition isn’t wrong.
However, just like sample mean is not distribution mean, sample average energy is not temperature. If we actually look at the math, the two are different. Sample average energy summarizes all the relevant information about temperature, but is not itself the temperature.
Of course, if we had perfect information about all the low-level particles, we might not have any need to use temperature to model the system. (In the same way, if we had perfect knowledge of all fish weights, we might not need to explicitly use a distribution to model them, depending on our use-case.)
One thing I didn’t explicitly mention in the post is that the average energy of the sample is a sufficient statistic for the temperature—it summarizes all the information from the sample relevant to the temperature. So in that sense, it is all we care about, and your intuition isn’t wrong.
However, just like sample mean is not distribution mean, sample average energy is not temperature. If we actually look at the math, the two are different. Sample average energy summarizes all the relevant information about temperature, but is not itself the temperature.
Of course, if we had perfect information about all the low-level particles, we might not have any need to use temperature to model the system. (In the same way, if we had perfect knowledge of all fish weights, we might not need to explicitly use a distribution to model them, depending on our use-case.)