In a variety of biological processes, ranging from branching morphogenesis and embryo implantation to cancer invasion, active tissues adhere to and penetrate the fibrous extracellular matrix (ECM). Although the engulfment morphologies often appear identical, the driving mechanisms that govern these processes can differ fundamentally. The tissue may engulf the ECM by deforming it through active tensions within the cellular tissue, akin to elastocapillarity, by degrading the substrate, or by employing a complex combination of both. Since the cellular fate and behavior are dictated by the mechanosensitive response to the mechanical state of the ECM, disentangling the implication of the competition between these two modes is crucial. This work investigates the competitive dynamics between mechanical indentation and degradation by modeling the cellular tissue as an active fluid droplet interacting with a hyperelastic solid substrate. We adopt a continuum model based on a Nitsche-Onsager variational formulation [Cicconofri et al. 2024]. To address the computational challenges of multi-phase flow and non-material moving boundaries, we developed a numerical framework utilizing non-conforming meshes and a phase-field description. We demonstrate the high predictive robustness of this approach through validation against analytical theories of wetting and experimental data on elastocapillarity effects [Style 2013] and human colon carcinoma (HCT-8) cell colonies [Shi et al. 2022]. Leveraging this framework, the study of active tissues reveals a fundamental mechanosensing inhibition mechanism in which the degradation rate is negatively regulated by substrate tension. Furthermore, relaxation simulations indicate that this interaction leaves a permanent invasive signature: a residual deformation field persisting after tissue removal.