Accurate tumor detection in digital pathology whole-slide images (WSIs) is crucial for cancer diagnosis and treatment planning. Multiple Instance Learning (MIL) has emerged as a widely used approach for weakly-supervised tumor detection with large-scale data without the need for manual annotations. However, traditional MIL methods often depend on classification tasks that require tumor-free cases as negative examples, which are challenging to obtain in real-world clinical workflows, especially for surgical resection specimens. We address this limitation by reformulating tumor detection as a regression task, estimating tumor percentages from WSIs, a clinically available target across multiple cancer types. In this paper, we provide an analysis of the proposed weakly-supervised regression framework by applying it to multiple organs, specimen types and clinical scenarios. We characterize the robustness of our framework to tumor percentage as a noisy regression target, and introduce a novel concept of amplification technique to improve tumor detection sensitivity when learning from small tumor regions. Finally, we provide interpretable insights into the model's predictions by analyzing visual attention and logit maps. Our code is available at https://github.com/DIAGNijmegen/tumor-percentage-mil-regression.