Why is concentration used as a proxy for activity instead of mole fraction?
The posted notation seems unorthodox if not incorrect.
For the Gibbs free energy of reaction one usually writes
$$\Delta_r G = \left(\frac{d \Delta G}{d \xi}\right) = \sum_i \nu_i\mu_i^\circ  + RT\sum_i  \nu_i \ln \gamma_i x_i$$
$$ = \Delta _r G^\circ + RT \ln Q$$
where $ \xi$ is the reaction progress, $Q=\sum_i  \nu_i \ln \gamma_i x_i$ is the reaction quotient, and one can equate $\Delta _r \mu= \sum_i  \nu_i\mu_i $ with $\Delta_r G$. For the reaction $A \leftrightharpoons B$ one would write $\Delta \mu = \mu_B - \mu_A$.
Changing concentration units is ok but a set of units usually implies a choice of thermodynamic reference state. The reference state generally defines the limit in which $\gamma \rightarrow 1$ (the limit of ideal behavior). If you decide to express the reaction quotient Q in terms of molar concentration rather than mole fraction but want to retain the same general form of the above equations, then it becomes necessary to modify both Q and $\Delta _r G^\circ$. Some reference states are better choices than others under certain circumstances, and it is not accurate to say that mole fraction is not used, it is, just not always. For solids or solvents a mole fraction unit is often convenient. When dealing with solutions, and particularly with solutes as a minor component, it is generally more convenient to deal in molal and molar units (the prior being favored), in combination with Henry's law. In addition, at small concentrations of solute the mole fraction, molarity and molality are linearly proportional, differing by a multiplicative constant.  For instance, at low concentration we can write the mole fraction of solute as
$$\chi _2 = \frac{n_2}{n_1 + \sum_i n_i} \approx \frac{n_2}{n_1} = \frac{c_2}{c_1} = \frac{c_2}{1000\delta_1/M_1} \approx \frac{M_1}{1000 \delta_1 ^\circ}c_2$$
where 1 and 2 label solvent and solute and $\delta_1 ^\circ$ is the mass density of the pure solvent, $M_1$ its molar mass.