These are scaling arguments, good for back-of-the-envelope calculations and for broadly classifying process regimes. The actual values may be uncertain by an order of magnitude of more. The important aspect is that $\Delta T$ and $10\Delta T$ and $\Delta T/2$ are all considered to be approximately equal and fundamentally different from $(\Delta T)^2$ or $\ln(\Delta T/T_0)$, for example.
Thus, is doesn't matter if the cross-sectional area is that of a rectangle ($L^2$) or a circle ($\pi L^2/4$), for instance; the important part is the $\sim\!\! L^2$ scaling.
Now to your question. $\Delta T$ is the temperature difference across the solid. Yes, this varies over time. Yes, an energy storage rate $\rho L^3c\Delta T/t$ is strictly meaningful only if the entire solid heats up by a temperature difference $\Delta T$, which is not the same as a temperature difference across the solid. In other words, a spatial gradient in temperature is being equated to a transient difference in temperature. For the reasons given above, this is acceptable for this level of analysis and further can provide insight not available otherwise. What we're checking is whether the Fourier number is, say, $10^{-5}$ or $1$ or $10^5$; it doesn't matter if some temperature estimate was inaccurate by a factor of two. Does this make sense?