From my understanding:
Variational autoencoders sample the latent variables $y$ using a proposal distribution $q$ of the observed variables $x$. The objective is that the decoder $p$ applied to $y$ defines a distribution whose value at $x$ is supposed to be maximized (reconstruction cost) plus a regularizer term.
Why not forget about $q$, and for every example $x$, iteratively update $y$ so as to maximize the objective? That is, setting $y$ to
$$\arg \max_y P(x | y)$$
rather than sampling it from $q(y \mid x)$.
Is $q$ an approximation motivated by inference speed?