Explainable Artificial Intelligence (XAI) is rapidly becoming an emerging and fast-growing research field; however, its adoption in healthcare is still at the early stage despite the potential that XAI can bring to the application of AI in this industry. Many challenges remain to be solved, including setting standards for explanations, the degree of interaction between different stakeholders and the models, the implementation of quality and performance metrics, the agreement on standards for safety and accountability, its integration into clinical workflows, and IT infrastructure. This paper has two objectives. The first one is to present summarized outcomes of a literature survey and highlight the state-of-the-art for explainability including gaps, challenges, and opportunities for XAI in healthcare industry. For easier comprehension and onboarding to this research field we suggest a synthesized taxonomy for categorizing explainability methods. The second objective is to ask the question if applying a novel way of looking at explainability problem space, through a specific problem/domain lens, and automating that approach in an AutoML similar fashion, would help mitigate the challenges mentioned above. In the literature there is a tendency to look at the explainability of AI from model-first lens, which puts concrete problems and domains aside. For example, the explainability of a patient's survival model is treated the same as explaining a hospital cost procedure calculation. With a well-identified problem/domain that XAI should be applied to, the scope is clear and well-defined, enabling us to (semi-) automatically find suitable models, optimize their parameters and their explanations, metrics, stakeholders, safety/accountability level, and suggest means of their integration into clinical workflow.