Missing entries in multi dimensional data pose significant challenges for downstream analysis across diverse real world applications. These data are naturally represented as tensors, and recent completion methods integrating global low rank priors with plug and play denoisers have demonstrated strong empirical performance. However, these approaches often rely on empirical convergence alone or unrealistic assumptions, such as deep denoisers acting as proximal operators of implicit regularizers, which generally does not hold. To address these limitations, we propose a novel tensor completion framework grounded in the monotone inclusion paradigm. Within this framework, deep denoisers are treated as general operators that require far fewer restrictions than in classical optimization based formulations. To better capture holistic structure, we further incorporate generalized low rank priors with weakly convex penalties. Building upon the Davis Yin splitting scheme, we develop the GTCTV DPC algorithm and rigorously establish its global convergence. Extensive experiments demonstrate that GTCTV DPC consistently outperforms existing methods in both quantitative metrics and visual quality, particularly at low sampling rates. For instance, at a sampling rate of 0.05 for multi dimensional image completion, GTCTV DPC achieves an average mean peak signal to noise ratio (MPSNR) that surpasses the second best method by 0.717 dB, and 0.649 dB for multi spectral images, and color videos, respectively.