Mean field theory models of percolation on networks provide analytic estimates of network robustness under node or edge removal. We introduce a new mean field theory model based on generating functions that includes information about the tree-likeness of each node's local neighbourhood. We show that our new model outperforms all other generating function models in prediction accuracy when testing their estimates on a wide range of real-world network data. We compare the new model's performance against the recently introduced message passing models and provide evidence that the standard version is also outperformed, while the `loopy' version is only outperformed on a targeted attack strategy. As we show, however, the computational complexity of our model implementation is much lower than that of message passing algorithms. We provide evidence that all discussed models are poor in predicting networks with highly modular structure with dispersed modules, which are also characterised by high mixing times, identifying this as a general limitation of percolation prediction models.