The title is pretty much my question. I haven't seen any literature yet that uses a different training objective. The goal is to find the hidden states eventually, then why is it that only 1 method is so popular, and there are no others seen?
Sequence prediction is not the only objective in RNNs and LSTMs.
For instance, we have sequence classification, where the input to the RNN/LSTM is the sequence, and the output is its label; a concrete example of that is sentiment analysis, where the input is a sentence, and the output is whether the answer is positive, negative or neutral. You can see specific models and benchmark datasets at paperswithcode.
Another objective is sequence tagging where, for each timestep, we output a label; an example is part-of-speech (POS) tagging, where the input is a textual sequence and the output is the POS tag each of the words in the text has. You can see specific models and benchmark datasets at paperswithcode.