"It would be interesting to know how different a model would be if it operated on eg dependency trees instead of the linear list of tokens." - indeed, this is obviously interesting, so people have tried that a lot for many models, but IMHO it's probably now almost decade since the consensus is that in general end-to-end training (once we became able to do it) work better than adding explicit stages in between, e.g. for any random task I would expect that doing text->syntax tree->outcome is going to get worse results than text->outcome, because even if the task really needs syntax, the syntax representation that a stack of large transformer layers learns implicitly tends to be a better representation of the natural language than any human linguist devised formal grammar, which inevitably has to mangle the actual languge to fit into neat human-analyzable 'boxes'/classes/types in which it doesn't really fit and all the fuzzy edge cases stick out. Once you remove the constraint that the grammar must be simplified enough for a human to be able to rationally analyze and understand, and (perhaps even more importantly?) abandoning the need to prematurely disambiguate the utterance to a single syntax tree instead of accepting that parts of it are ambiguous (but not equally likely), processing works better.
It's just one more reminder of the bitter lesson (http://incompleteideas.net/IncIdeas/BitterLesson.html) which we don't want to accept.