The difference between an approximation of acceleration and an exact formulation of average acceleration can be seen by observing that thee data points are sufficient to approximate instantaneous acceleration; a formulation of average acceleration requires a minimum of four data points.
That I have provided distinct expressions for these distinct quantities should be sufficient to convince the qualified reader of the distinction between these similar concepts.
The answer is: Yes, it is possible to formulate an exact expression which can reasonably be called average acceleration. What I have called the approximate acceleration at time $t_p$ is an exactly defined value derived from two average velocities. Applying the same method to two consecutive "approximate" accelerations gives the two forms
$$\left\langle \mathfrak{a}_{p}\right\rangle \equiv\frac{\mathfrak{a}_{p}-\mathfrak{a}_{p-1}}{\Delta t}$$
$$=\frac{1}{\Delta t^{2}}\left(\left\langle \mathfrak{v}_{p+1}\right\rangle -2\left\langle \mathfrak{v}_{p}\right\rangle +\left\langle \mathfrak{v}_{p-1}\right\rangle \right)$$
$$=\frac{1}{\Delta t^{3}}\left(\mathfrak{r}_{p+1}+3\left(\mathfrak{r}_{p-1}-\mathfrak{r}_{p}\right)-\mathfrak{r}_{p-2}\right).$$
I'm not convinced of the utility of this expression, but it is an exact formulation consisting entirely of the input data, having a form similar to that used for average velocity.
I strongly urge anyone wishing to borrow this result to double-check my algebra and reasoning.