The rapid development of the low-altitude economy (LAE) has created growing demand for reliable aerial communication systems. Extremely large-scale multiple-input multiple-output (XL-MIMO) is a promising enabler for such systems due to its high spatial resolution and robust connectivity. However, three-dimensional (3D) mobility together with near-field propagation makes it difficult to obtain dedicated high-fidelity wireless datasets, hindering systematic algorithm development and evaluation. To address this issue, we develop LAETwin-XL, a digital twin (DT)-based toolchain and dataset for XL-MIMO research in LAE scenarios. Built on the Sionna ray-tracing (RT) module, the proposed toolchain simulates near-field and far-field channels with diverse wireless labels for practical environments. Building on this dataset, we further develop a conditional denoising diffusion implicit model (CDDIM)-based generative foundation model that is pretrained to learn transferable XL-MIMO channel representations from incomplete channel observations. Unlike conventional task-specific or foundation models that rely on relatively complete channel inputs, the proposed model can generatively infer informative channel representations from partially observed channels. Experimental results demonstrate that the proposed framework achieves effective zero-shot channel extrapolation performance. Furthermore, using lightweight task heads and limited training data, it enables parameter-efficient transfer to various downstream tasks (e.g., channel estimation, classification, and localization), delivering high accuracy and robustness even under sparse antenna observations. The codes and dataset are available at https://github.com/Lmyxxn/LAETwin-XL.