Why does specific heat capacity affect thermal diffusivity?

When you heat an object the way you say you are saying, the object does not heat uniformly, and the temperature gradient within the object is not uniform. The heat first begins getting stored in the side of the object you are applying the heat, penetrating only a small distance in. As time progresses, owing to the high temperature gradient near the side you are heating, and the lower temperature gradient further in, the heat penetrates further into the object. The shc determines how much can get stored in this thermal boundary layer, and, overall, the thermal diffusivity determines how far the penetration is at a given time.

Let's discretize time and space so we can apply somewhat simpler equations than the spatiotemporal partial differential equation $alphadot{T}=T^{primeprime}$. Your example of a high-$c_P$ object and a low-$c_P$ object being heated is a great starting point; $c_P$ is of course the specific heat capacity, and let's assume the heating flux $q$ is applied in the $x$ direction.

Time step 1: In a small volume at the surface—call it element 1—the temperature (relative to the ambient temperature) increases from zero to $$Delta T=qADelta t/mc_P,$$ where $A$ is the element cross-sectional area moving inward, $Delta t$ is the time step duration, and m is the element mass.

(At this first time step, there's not yet any conduction inward from element 1 because this discretization scheme has us looking at the element temperatures at the beginning of the time step; the material starts out at a uniform relative temperature of zero.)

Clearly, the high-$c_P$ material heats up less.

Time step 2: Let's now analyze the inward heating by applying Fourier's law of conduction to element 1 and identical element 2 just inward: $$q_mathrm{cond}=kDelta T/L=qkADelta t/mc_PL,$$ where $k$ is the thermal conductivity and $L$ is the element depth moving inward.

We find that the inward heat flux is lower in the high-$c_P$ material, as all other variables are equal.

Additional time steps: Because of the lower incoming flux, the next block inward heats up less in the high-$c_P$ material, and the corresponding conduction through its inner side is subsequently lower. Analysis of $n$ subsequent time steps produces terms that scale with $c_P^{-n}$, emphasizing the importance of the specific heat. Equivalently, the heat wave has been effectively delayed and the speed of a given temperature change reduced in the high-$c_P$ material, as predicted by the lower thermal diffusivity.

So when you say, "It['s] not like moving heat into an object with a high SHC just 'destroys' that energy", you have a point—energy is never destroyed—but it's nevertheless true that the temperature increase in the material is suppressed, as is the propagating energy flux $q_mathrm{cond}$ that's driven by temperature gradients. In this sense, a higher-$c_P$ material both absorbs more energy and delays heat transfer.