It's an interesting idea. Some miscellaneous thoughts:
If you allow arbitrary mappings from data space to representation, then it seems that your method is more flexible than autoencoders, and includes autoencoders as a special case (i.e. when the mapping exactly matches the encoder portion of an autoencoder). This means that, when an autoencoder can solve the problem, there exists a configuration of your method that can also solve the problem (but doesn't guarantee that this configuration is learnable). It might also be the case that some configurations of your method can solve problems that autoencoders cannot, with the same caveat.
If you have $n$ data points and your latent space has dimensionality $d$, then you'd have $nd$ parameters for the latent representations, plus the parameters of the decoder network. If your original data has dimensionality $d'$, an autoencoder would have $d d'$ parameters for the encoder, plus parameters for the decoder. In the case where you have many more data points than dimensions, your approach may be much more flexible than an autoencoder. This could be beneficial, but also means you might have to worry more about overfitting and possibly runtime. Unlike an autoencoder, it's not obvious that you could escape overfitting by adding more data, since the number of parameters scales with number of data points. It seems like constraints from the 'bottleneck'/decoder would have to be the thing that saves it.
Your optimization problem will most likely be nonconvex, and SGD will converge to a local minimum. My gut feeling is that the results might be highly dependent on the initial configuration, particularly of the initial latent representations.
Compared to autoencoders, your method won't give a mapping from the data space to the latent space, so it can't directly perform out-of-sample generalization (although this may not be needed, depending on your goal). This is common to many nonlinear dimensionality reduction algorithms. If necessary, you could use some auxilliary method to learn such a mapping. It does mean you'd have to think about how to perform validation (e.g. how would you test your algorithm on a held-out data set?).
There's certainly precedent for treating the latent representations of every data point as parameters, and directly optimizing them. Nonclassical multidimensional scaling (MDS) is one example. The objective function is nonconvex, and points are iteratively repositioned in the latent space to find a local minimum. It often helps to initialize with classical MDS (which is convex and can be solved in one shot using an eigendecomposition). Many nonlinear dimensionality reduction methods also treat the latent representations of every point as parameters.