> My conclusion is then that entropy is not a property of physical systems themselves, but rather a property of the way we choose to study them.

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.

For example, in thermodynamics, in some practical cases, we can define what we mean by work and heat quite well. For a continuous change of macrostate from some reference state $\mathbf X_0$ to a final state $\mathbf X$, if this can be made reversibly (no violent processes allowed) over some path $\gamma$ in the space of macrostates, joining $\mathbf X_0$ to $\mathbf X$, we can define function of the final state $\mathbf X$:
$$
S(\mathbf X) = \int_{\gamma}\frac{dU - dW}{T}.
$$
It turns out that provided 2nd law holds, this quantity has interesting properties, such as being a function of  the macrostates $\mathbf X_0, \mathbf X$ only (it does not depend on details of the path $\gamma$). So it is a function of the state, independent of the path we used to get there, just like internal energy $U$ is. Clausius and others used this to formulate thermodynamic laws; e.g. reversible processes in closed, thermally insulated systems which are allowed to exchange work, are characterized by entropy remaining constant; and irreversible processes in closed thermally insulated systems are characterized by entropy increasing.

> But then entropy is not as fundamental a concept as it is usually made out to be - it is simply a mathematical consequence of a choice we make, i.e. which parameterizations of the system we will compare. So why do we bother studying entropy if it is a purely mathematical phenomenon?

I think largely for historical reasons, because it is hard to understand, and because it was written about incorrectly by previous authors, so it provokes people to write about it and then other people write why that is wrong. 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 did not catch on; in a sense, entropy has won. It's like an enigma and buzzword wrapped into a science project, 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.

With development of statistical physics and theory of communication (information theory), people introduced new and different concepts and formulae with similar behavior to Clausius entropy (Boltzmann, Gibbs), and later entirely *other* people named those *entropy* too, even though the concepts are different(Boltzmann entropy, Gibbs entropy). This definitely confused and continues to confuse a lot of people.

Today, we have also the Shannon entropy, and in quantum theory, the von Neumann entropy, and there are even more modern contributions, such as the Kolmogorov entropy or the Renyi entropy.

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, and people write about entropy in a way the other people do not agree with.