In cases such as Gaussian mixture models, there's is no closed-term solution for the original likelihood maximization. Maximizing the ELBO, however, does have analytical update formulas (i.e. formulas for the E and M steps). I understand why in this case maximizing the ELBO is a useful approximation.
However, in more complex models, such as VAE, the E & M steps themselves don't have a closed solution, and ELBO maximization is done with SGD. In this scenario, what's the advantage of optimizing the ELBO with SGD over maximizing the original likelihood with SGD?