Callen's condition of homogeneity of degree one for the entropy as a function of its extensive variables is a direct consequence of his third postulate.
One has to apply the postulate to a compound system made by $n$ equal systems. The fact that they are equal means that they are characterized by the same values of temperature, pressure, and chemical potential. Therefore, if we would remove the walls between them, we will remain with a system still at equilibrium characterized by an energy that is $n$ times the energy of a subsystem, and similarly for volume and number of particles (or moles). In the formula, in such a case, due to the equilibrium condition, it is true that
$$
S(nU,nV,nN)= n S(U,V,N) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [1]
$$
for all positive integer values of $n$ and for all ($U,V,N$) in the domain of $S$.
Thus, let's introduce $\tilde U= nU,\tilde V = nV,\tilde N = nN$. Equation [$1]$ becomes
$$
S(\tilde U,\tilde V,\tilde N)= n S\left(\frac{\tilde U}{n},\frac{\tilde V}{n},\frac{\tilde N}{n} \right) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [2]
$$
i.e., taking into account the arbitrariness of the arguments
$$
S\left(\frac{U}{m},\frac{ V}{m},\frac{ N}{m} \right) = \frac{1}{m}S( U, V, N),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [2^{\prime}]
$$
again for all positive integer values of $m$ and for all ($U,V,N$) In the domain of $S$.
Combining [$1$] and [$2^{\prime}$] we get
$$
S\left(\frac{nU}{m},\frac{n V}{m},\frac{n N}{m} \right) = \frac{n}{m}S( U, V, N),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [2'']
$$
for all integer positive values of $n$ and $m$ and for all ($U,V,N$) in the domain of $S$.
At this point, it is enough to work with sequences of ratios $n_i/m_i$ and using the continuity of entropy (again in the third postulate) to prove homogeneity of degree one
$$
S\left(\lambda U, \lambda V,\lambda N \right) = \lambda S( U, V, N)
$$
for all real positive values of $\lambda$ and for all ($U,V,N$) in the domain of $S$.
As you can see, there is no mistake since the total system and subsystems are all at mutual equilibrium.
A word of caution is required for the domain of $S$. Callen does not discuss it, but it is clear that homogeneity is meaningful only if $U, V,$ and $N$ are positive. That is not a problem for $V$ and $N$. However, also $U$ is not an issue if we require that internal energy is limited below. Since it is always possible to add an arbitrary constant to energy, we can take the lower limit as the new zero of energy. Similarly for the entropy, which is limited below as a consequence of the fourth principle of thermodynamics.
As a final remark, I would add that I have played for a while with alternative formulations of Callen's postulates. My partial conclusion is that Callen's choice of postulates looks wise and carefully planned.
