So, doesn't specific heat capacity of a material affect the rate of heat transfer/flux ?
It does: just look at Fourier's equation of heat conduction:
$$\frac{\partial T}{\partial t}=\alpha \Big(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\Big)$$ where $T$ is the temperature and: $$\alpha=\frac{k}{\rho c_p}$$
is the thermal diffusivity of the conducting material. $k$ is the heat conductivity and $c_p$ is the heat capacityheat capacity of that material.
I'll illustrate how this equation works with a simple $1D$ example. A uniform rod of length $L$ is clamped with one end at $T_1$ and one end at $T_2$. The rod is insulated along its length. What's the temperature evolution $T(x,t)$ of the rod?
The solution is$^{\dagger}$ given by:
$$\begin{align*}T\left( {x,t} \right) & = {T_1} + \frac{{{T_2} - {T_1}}}{L}x + \sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right){{\bf{e}}^{ - \alpha{{\left( {\frac{{n\pi }}{L}} \right)}^2}\,t}}} \end{align*}$$
The thermal diffusivity $\alpha$ appears in the time-based exponential decay factors:
$$e^{-\alpha\Big(\frac{n\pi}{L}\Big)^2t}$$
So the higher the value of $\alpha$, the faster that function decays to $\text{zero}$, which is for lower values of $c_p$.
$^{\dagger}$ note that the author confusingly used $k$ where it should be $\alpha$.