You're being loose with semantics and drawing bad conclusions. The training describes the nature of the feedback given to the system. Being trained to predict tokens means it gets feedback on the quality of its predictions. This does not constrain or determine the structure of the learned model. The learned model is a function of the training time, quality and amount of data, as well as the architecture that is biased towards capturing certain features or entails limits on what can be captured. The point is, the space and properties of solutions to the problem of "predicting the next token" is not constrained by the nature of the feedback.
It absolutely does! A human can learn to perform an action by reading a manual. But an LLM is only trained to predict tokens, if you feed it a manual it will learn to recite the manual, it wont learn how to perform the actions in the manual.
This makes the training inherently flawed from a learning perspective, and that flaw is impossible to fix without moving past the token prediction way of training models.
This claim doesn't track with LLM-based agents. For example: https://towardsai.net/p/machine-learning/meet-webagent-deepm...
The method signature at the end doesn't matter you are right there but what the model is trained to do matters a ton. If it was trained to solve logical problems or to fact check etc, then it is no longer just a token predictor. But as is all the model is trained to do is to predict tokens, and you notice that immediately when you use it because it leads to obvious issues with what the model can do. It is really easy to target sparse parts of its data and get it to say nonsense because it is just trained to predict tokens.