Why can't this system be adequately represented using a z-domain transfer function?
A transfer function describes an LTI system. As such, the given system can be described by a transfer function. However, if there are non-zero initial conditions, the system is no longer linear because there's a contribution in the output that does not depend on the input signal but only on the initial conditions. Consequently, the transfer function cannot directly be used to compute the system's response if there are non-zero initial conditions.
Nevertheless, the (unilateral) $\mathcal{Z}$-transform can still be used to compute the system's response, even with non-zero initial conditions by transforming the difference equation and using
$$\mathcal{Z}\big\{ y[n-k]\big\}=z^{-k}Y(z)+\sum_{m=0}^{k-1}z^{-m}y[m-k],\qquad k&gt;0\tag{1}$$
EXAMPLE:
Let's use a simple example with a similar pole-zero cancellation as in the original problem to illustrate the point. Consider a system described by
$$y[n]-y[n-1]=\alpha \big(x[n]-x[n-1]\big)\tag{2}$$
The corresponding transfer function is
$$H(z)=\frac{Y(z)}{X(z)}=\frac{\alpha(1-z^{-1})}{1-z^{-1}}=\alpha\tag{3}$$
Clearly, $y[n]=\alpha x[n]$ is a solution of $(2)$. It is also the only solution if we require the system to be linear. However, it is not the only solution if we allow non-linear systems because there are infinitely many solutions of the form
$$y[n]=\alpha x[n]+c\tag{4}$$
with an arbitrary constant $c$. Note that these solutions cannot be inferred from the transfer function $(3)$.
Let's now use the $\mathcal{Z}$-transform to solve $(2)$ with initial conditions $y[-1]\neq 0$ and $x[-1]=0$. Transforming $(2)$ using $(1)$ gives
$$Y(z)(1-z^{-1})-y[-1]=\alpha X(z)(1-z^{-1})$$
which results in the following $\mathcal{Z}$-transform of the output:
$$Y(z)=\alpha X(z)+\frac{y[-1]}{1-z^{-1}}\tag{5}$$
In the time domain this becomes
$$y[n]=\alpha x[n]+y[-1]u[n]\tag{6}$$
where $u[n]$ is the unit step. Eq. $(6)$ is just a causal version of $(4)$.
This shows that the $\mathcal{Z}$-transform can be used to compute the system's response with non-zero initial conditions, even though the transfer function alone is inadequate for solving the problem.