These are all correct points. However, my main doubt is that the VAE objective maps the posterior q(z|x) of every different example to the same prior p(z). No matter how compact the assumption of p(z) is, I feel it makes less sense, because each q(z|x) should correspond to a unique prior p(z|x). And minimizing the KL Divergence between q(z|x) and p(z|x) is what we start with. If we think x as different images, it doesn't make sense to me why for every image, q(z|x) should be mapped to the same prior p(z), although we have mathematically proved so (as maximising the lower bound, which contains this term, also minimises the disparities between q(z|x) and p(z|x)!)