After reading these posts: Why is the partition function divided by $(h^{3N} N!)$?, What is the resolution to Gibb's paradox?, and these articles: The Gibbs Paradox and the Distinguishability of Identical Particles , The Gibbs Paradox, I understand that the division of the partition function by $N!$ has no relationship with the fact of identical particles.
Thus, I ask: If the partition function is given as a normalization factor for the probability of a system, that is: $$p_r = {e^{- \beta \epsilon_r} \over Z}$$ where $Z=\sum_r e^{- \beta \epsilon_r} $, I have read that the total partition function of the system is $$Z_\text{tot} = Z^N$$ where $N$ the total number of particles, so we can say that the total average energy of the system is $$\langle E\rangle=N \epsilon_1$$ with $$\epsilon_1 =-kT {\partial (\ln Z) \over \partial \beta} $$
Why don't we divide the total partition function $Z_\text{tot}$ by $N!$, if this division has no relationship with the fact that in quantum mechanics the particles are identical? Then we should have that $$Z_\text{tot}={Z^N \over N!}$$ So why don't we divide by $N!$?
Note: The references at the top of the posts are about the division by the factorial. As I have understood, they argue that the division by $N!$ is not because the particles are identical but because we want the thermodynamic entropy to be the same as the statistical entropy (a matter of convenience). So why isn't this division necessary here?
Note: After a first answer I am making a note here: If someone argues that the division happens because of the Gibbs paradox, they should at least present a counter-answer to the references I am making at the top of the post — arguing at least why those posts and articles contain errors or are entirely mistaken and, if possible, provide more sources for reading and study.
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