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I think largely for historical reasons; because it was(is) hard to understand, and because it was written about incorrectly by some authors, it provoked people to write about it and then other people write why that is wrong. So it populated scientific articles and textbooks. Clausius invented thermodynamic entropy, formulated thermodynamics with it, and then people started to analyze what it means (and they do not seem to be done, especially when connection to statistical physics is studied, or when deciding how the concept should be taught). This created a lot of confusion, even resistance. I remember reading about a scientist who rejected entropy as a concept to use when formulating thermodynamics, and formulated everything without it. This is possible, but did not catch on; in a sense, entropy has won societally. It's like a scientific enigma to be decrypted, people accept the trouble with it because they think some big insight or development of our understanding is possible. Clausius and others started this mysticism with statements like "energy of universe is constant, entropy of universe only ever increases" or something like that, which I think is not scientific, more like a catchy slogan to promote one's ideas or maybe a first instance of a buzzword-fueled marketing.

The set of macroscopic variables $\mathbf X$ is what we choose to describe the system. For homogeneous gas, we can use $U,V$, but if we have a reason to include external magnetic field $B_{ext}$, we can. In the future, if we discover a new way to interact with the gas state, e.g. via some new field $G$, then entropy function will change to be a function of $G$ as well.

Yes, the entropy function arguments and its value depend on the choice of variables. However, in case of the Clausius entropy, changes of this entropy between two states are related to integral of $dQ/T$, which are definite measurable quantities, so changes of this entropy do not depend on the choice of variables $\mathbf X$.

I think we study it largely for historical reasons; because it was(is) hard to understand, and because it was written about incorrectly by some authors, it provoked people to write about it and then other people write why that is wrong. So it populated scientific articles and textbooks. Clausius invented thermodynamic entropy, formulated thermodynamics with it, and then people started to analyze what it means (and they do not seem to be done, especially when connection to statistical physics is studied, or when deciding how the concept should be taught). This created a lot of confusion, even resistance. I remember reading about a scientist who rejected entropy as a concept to use when formulating thermodynamics, and formulated everything without it. This is possible, but did not catch on; in a sense, entropy has won societally. It's like a scientific enigma to be decrypted, people accept the trouble with it because they think some big insight or development of our understanding is possible. Clausius and others started this mysticism with statements like "energy of universe is constant, entropy of universe only ever increases" or something like that, which I think is not scientific, more like a catchy slogan to promote one's ideas or maybe a first instance of a buzzword-fueled marketing.

Almost correct. However, some of these ways we choose to study are well established and well-defined, and then entropy is well-defined and (to some) interesting.

Almost correct. However, some of these ways we choose to study are well established and well-defined, and then a particular well-defined function can get widely accepted and (to some) is interesting.

All of these can be indicted with some anthropomorphism, or dependence on some human choice; we choose variables we consider a "practical description" of the physical state, either real physical quantities such as $U,V$, or probabilities of microstates $p_i$, and then various entropies are functions of these preferred variables. We could have chosen the variables differently, but if there is no good reason, we use the smallest number of the most plausible variables possible, which (hopefully) makes everyone agree on the one entropy function to be used. Sometimes this fails, or people are unclear which entropy function they talk about, or use it as if it did not depend on those choices (Landauer), and sometimes other people have issue with that.

All of these can be indicted with some anthropomorphism, or dependence on some human choice; we choose variables we consider a "practical simplified description" of the physical state, either real physical quantities such as $U,V$, or probabilities of microstates $p_i$, and then various entropies are functions of these or other preferred variables. We could have chosen the variables differently, but if there is no good reason, we use the smallest number of the most plausible variables possible, which (hopefully) makes everyone agree on the one entropy function to be used. Sometimes this fails, or people are unclear which entropy function they talk about, or use it as if it did not depend on those choices (Landauer), and sometimes other people have issue with that.