In this work we expand the previously proposed Integrated Finite Element Neural Network (IFENN) framework to simulate the complete evolution of cracks within a phase-field formulation, encompassing both crack initiation and propagation. IFENN is a hybrid computational methodology that embeds pre-trained neural networks into a nonlinear FEM solver to accelerate the solution of multi-physics problems. The mechanical equilibrium is resolved using conventional FEM, whereas the remaining physics-based equations are approximated by a pre-trained neural network (NN). The key feature of the framework is the iterative exchange of information between the two solvers, akin to a staggered scheme, which avoids blind full-field predictions and ensures seamless coupling between the two numerical methods. Here we showcase how IFENN can be implemented to model the particularly challenging problem of phase-field crack evolution, involving both the initiation (pre-peak) and propagation (post-peak) stages. For the first stage the neural network of choice is a Deep Operator architecture with Kolmogorov Arnold Networks (KANs) for the branch and the trunk, while crack propagation is modelled with the aid of Convolution Neural Networks (CNNs). Both networks are trained exclusively on the physical relationship between the maximum strain energy density and the phase-field variable, utilizing very few training increments and avoiding expensive data generation. We note that KANs naturally accommodate unstructured data, whereas, during the propagation stage, this capability is achieved through a fast interpolation step that maps unstructured finite element data onto a structured grid suitable for CNN inference. The proposed framework is implemented across several benchmark problems, readily demonstrating the accuracy, computational speedup, and generalizability across unseen test cases.