Why do we approximate the joint in ELBO if we already have access to it? I realized in variational inference, our goal is to approximate $p(z|x)$ with $q(z)$. So we minimize $KL(q(z) || p(z|x)) = \mathbb{E}_{z \sim q} log\frac{q(z)}{p(z|x)}$. We then manipulate, through some simple steps, to arrive at:
$$ = \mathbb{E}_{z \sim q} \left[ log \frac{q(z)}{p(z,x)}\right] + logp(x)$$
and then the objective becomes optimizing $q$ such that it is close to the joint... but don't we already have access to the full functional form of the joint? What's the point of approximating the joint if we already have access to it?
TLDR We want to minimize $KL(q(z) || p(z|x)) = \mathbb{E}_{z \sim q} log\frac{q(z)}{p(z|x)}$ but we don't have access to the posterior so we can't. We then decide to minimize $KL(q(z) || p(z,x)) = \mathbb{E}_{z \sim q} log\frac{q(z)}{p(z,x)}$ since we have access to the joint. My question, is why even approximate if we already have the joint?
 A: As you stated, we have:
$$
KL(q(z) || p(z|x)) = \mathbb{E}_{z \sim q} \left[ \log \frac{q(z)}{p(z,x)}\right] + \log p(x),
$$
where $-\mathbb{E}_{z \sim q} \left[ \log \frac{q(z)}{p(z,x)}\right]$ is also called the ELBO. And the goal is to minimize $KL(q(z) || p(z|x))$. And you are right, since the log evidence is constant, minimizing the RHS means maximizing the ELBO, which in turn means approximating $p(x, z)$ with $q(z)$.
But note that, although we do know the functional form of the joint $p(x, z) = p_\theta(x, z)$, we usually don't know the value for its parameter $\theta$. The same goes for $q(z)$: the functional form for $q(z) = q_\phi(z)$ is usually known (because it is chosen to be simple), but the parameter $\phi$ is not known. So the task is to find the parameters $\theta$ and $\phi$ by maximizing the ELBO.
E.g., think of some state space model, where you want to use variational inference to obtain the posterior of the state space $z$ given the observations $x$. You know that the state space model is defined by some parameters $\theta$ but those are not known to you. Next, you presume some function $q_\phi(z|x)$ that should approximate $p_\theta(z|x)$. This $q_\phi$, too, depends on some yet unknown parameters $\phi$ that need to be inferred (otherwise there would be nothing left to do since you then already know your approximation $q$). Another popular example would be the VAE where the parameters of the densities $p_\theta(x|z)$ and $q_\phi(z|x)$ like e.g. mean and covariance, are given by (deep) neural networks of prescribed architecture and the parameters $\theta$ and $\phi$ are the weights of the neural networks that need to be learned.
