Solving non-convex regularized inverse problems is challenging due to their complex optimization landscapes and multiple local minima. However, these models remain widely studied as they often yield high-quality, task-oriented solutions, particularly in medical imaging, where the goal is to enhance clinically relevant features rather than merely minimizing global error. We propose incDG, a hybrid framework that integrates deep learning with incremental model-based optimization to efficiently approximate the $\ell_0$-optimal solution of imaging inverse problems. Built on the Deep Guess strategy, incDG exploits a deep neural network to generate effective initializations for a non-convex variational solver, which refines the reconstruction through regularized incremental iterations. This design combines the efficiency of Artificial Intelligence (AI) tools with the theoretical guarantees of model-based optimization, ensuring robustness and stability. We validate incDG on TpV-regularized optimization tasks, demonstrating its effectiveness in medical image deblurring and tomographic reconstruction across diverse datasets, including synthetic images, brain CT slices, and chest-abdomen scans. Results show that incDG outperforms both conventional iterative solvers and deep learning-based methods, achieving superior accuracy and stability. Moreover, we confirm that training incDG without ground truth does not significantly degrade performance, making it a practical and powerful tool for solving non-convex inverse problems in imaging and beyond.