But it still seems to me that in some weird sense the wavefunction contains 'too much' information, in the same way that simulating a classically indeterministic universe by modelling all of its branches gives you 'too much information'.

I know what you mean, but as I'm sure you know it's not mere perversity that has led many to accept "modeling all the branches" of the QM universe. In the case of a classically indeterministic universe, you can model just one indeterministic branch, but in the case of the QM universe you can't do that, or you can't do it anywhere near as satisfactorily. The "weirdness" of QM is precisely that aspect of it which (in the eyes of many) forces us to accept the reality of all the branches.